KSQ-004EX Treatment for Advanced Solid Tumors

Inclusion Criteria: 1. Diagnosed with one of the following tumor types: 1. Unresectable, incurable and/or metastatic histologically and/or cytologically confirmed cutaneous, acral, or unknown primary melanoma (Stage IIIC or Stage IV) that has progressed following at least 1 and no more than 3 lines of prior therapy in the advanced/metastatic setting, one of which includes treatment with anti-PD-1/PD-L1 inhibitor alone or in combination with anti-cytotoxic T lymphocyte associated protein 4 (anti-CTLA-4) inhibitor or in combination with anti-LAG-3 antibody. Note: Up to 5 mucosal patients can be treated in the melanoma expansion cohort only. 2. Histologically and/or cytologically confirmed primary diagnosis of NSCLC which has progressed on at least 1 line and no more than 4 lines of prior therapy in the advanced/metastatic setting, including platinum-based chemotherapy and checkpoint inhibitor therapy (either given in combination or sequentially). i. Participants with tumors that have known actionable molecular alteration such as epidermal growth factor receptor (EGFR), anaplastic lymphoma kinase (ALK), ROS-1, B-Raf proto-oncogene (BRAF), rearranged during transfection (RET), MET and Kirsten Rat Sarcoma Virus (KRAS) must have progressed on standard directed molecular therapy in addition to platinum-based chemotherapy. Note, the following do not count towards a line of prior therapy: * Any therapy/regimen discontinued due to intolerance/tolerability issues * Retreatment with the same class or previous class of treatment alone or in combination c. Locally advanced recurrent and/or metastatic histologically and/or cytologically confirmed HNSCC that has been previously treated with at least 1 and no more than 4 lines of prior therapy in the advanced/metastatic setting. i. Participants must have received a platinum-containing chemotherapy regimen for the treatment of primary tumor in locally advanced, or metastatic setting d. Advanced, metastatic histologically and/or cytologically confirmed colorectal adenocarcinoma that has progressed following at least 1 and no more than 3 lines of prior therapy. Participants with dMMR/MSI-H or KRASG12C BRAF V600E mutation must have progressed on standard directed therapy. e. Locally advanced, recurrent, or metastatic histologically and/or cytologically confirmed pancreatic ductal adenocarcinoma (PDAC) that has progressed following at least 1 and no more than 3 lines of prior therapy in the advanced/metastatic setting. f. Recurrent, metastatic, or persistent histologically and/or cytologically confirmed squamous cell carcinoma (SCC), adenosquamous carcinoma, or adenocarcinoma of the cervix that is not amenable to curative treatment with surgery and/or radiation therapy that has progressed following at least 1 and no more than 3 lines of prior therapy in the advanced/metastatic setting. 2. Resectable lesion for KSQ-004EX manufacturing (tumor ≥ 1.5 cm2 or at least 5 core needle biopsies) 3. At least one measurable lesion per RECIST v1.1 (Eisenhauer 2009) following tumor resection for KSQ-004EX manufacturing Note: Lesions in previously irradiated areas should not be selected as a target lesion unless radiation treatment was ≥ 3 months prior, and there has been demonstrated disease progression in the lesion 4. Age ≥ 18 years of age 5. Life expectancy of ≥ 12 weeks 6. Recovered to ≤ Grade 1 or Baseline toxicity (except alopecia, neuropathy, and endocrinopathies from prior immunotherapy) from prior therapy (per the National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\]) 7. Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 8. Adequate bone marrow function defined as: 1. Absolute neutrophil count (ANC) of ≥ 1 × 109/L 2. Platelet count of ≥ 100.0 × 109/L 3. Hemoglobin of ≥ 9.0 g/dL 9. Adequate renal function defined as calculated creatinine clearance (Cockcroft-Gault) ≥ 40 mL/min 10. Adequate hepatic function defined as: 1. Total bilirubin ≤ 2.0 × upper limit of normal (ULN) unless associated with Gilbert's syndrome 2. Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3.0 × ULN (or ≤ 5 × ULN in patients with liver metastases) 11. Washout period from prior anticancer therapy(ies) of a minimum duration (excluding bridging therapy per concomitant medication, see Section 6.11) is required prior to the first study treatment (i.e., start of LDC therapy) as detailed below: 1. Targeted therapy: prior targeted therapy with an EGFR, ALK, receptor tyrosine kinase, or other-directed agent (eg, erlotinib, afatinib, osimertinib, crizotinib, ceritinib), is allowed provided the washout is a minimum of 7 days or 5 half-lives, whichever is longer prior to start of therapy (LDC) 2. Monoclonal antibodies with a washout of at least 21 days or 5 half-lives or whichever is longer 3. Chemotherapy: adjuvant, neoadjuvant or definitive chemotherapy/chemoradiation is allowed provided the washout is a minimum of 21 days or 5 half-lives, whichever is shorter 4. Radiation therapy: prior external beam radiation is allowed provided a minimum of 14 days have elapsed between the last dose of radiation and first study treatment (LDC) 5. Surgery: previous surgical procedure(s) is permitted provided that wound healing has occurred and at least 14 days have elapsed (for major operative procedures) prior to the first study treatment (LDC) 12. Female participants who are women of childbearing potential (WOCP), (defined as physiologically and anatomically capable of becoming pregnant), confirmed of a negative pregnancy test and agreement to the use of a highly effective contraceptive method or at least 2 effective methods at the same time during study treatment period and for up to 3 months after study treatment (KSQ-004EX infusion). Male participants must be willing to use effective barrier contraception (ie, condoms) during the study treatment period and for up 3 months after study treatment (KSQ-004EX infusion) a. This includes all female participants, between the onset of menses (as early as 8 years of age) and 55 years unless the participant presents with an applicable exclusionary factor which may be one of the following: i. Postmenopausal (no menses in greater than or equal to 12 consecutive months). ii. History of hysterectomy or bilateral salpingo-oophorectomy. iii. Ovarian failure (follicle stimulating hormone and estradiol in menopausal range, who have received whole pelvic radiation therapy). iv. History of bilateral tubal ligation or another surgical sterilization procedure. b. Approved methods of birth control are as follows: hormonal contraception (ie, birth control pills, injection, implant, transdermal patch, vaginal ring), intrauterine device (IUD), tubal Ligation or hysterectomy, subject/partner post-vasectomy, implantable or injectable contraceptives, and condoms plus spermicide. Not engaging in sexual activity for the total duration of the trial and the drug washout period is an acceptable practice; however periodic abstinence, the rhythm method, and the withdrawal method are not acceptable methods of birth control. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. 13. Ability to understand and the willingness to sign a written informed consent document 14. Signed and dated Institutional Review Board (IRB) approved informed consent form (ICF) before any protocol-directed Screening procedures are performed Exclusion Criteria: 1. Prior organ allograft or prior cell therapy that included LDC or myeloablative chemotherapy regimen 2. Known hypersensitivity to any component of KSQ-004EX or excipient including dimethyl sulfoxide, human serum albumin, LDC regimen (cyclophosphamide or fludarabine) or IL-2 (as applicable) 3. Active or prior documented autoimmune or inflammatory disorders (including inflammatory bowel disease \[eg, Grade ≥ 2 colitis or Crohn's disease\], systemic lupus erythematosus, sarcoidosis syndrome, or Wegener syndrome \[granulomatosis with polyangiitis\], rheumatoid arthritis, etc.\]). The following are exceptions to this criterion: 1. Participants with vitiligo or alopecia 2. Participants with hypothyroidism (eg, following Hashimoto syndrome) stable on hormone replacement 3. Any chronic skin condition that does not require systemic therapy 4. Participants with celiac disease controlled by diet alone 4. Hypersensitivity to antibiotics of the aminoglycoside group (eg, streptomycin, gentamicin) or penicillin 5. Active, uncontrolled concurrent infection requiring IV antibiotics present at Screening 6. Uveal and/or ocular melanoma 7. Large cell neuroendocrine NSCLC (defined as pathology with \>10% neuroendocrine components) 8. Symptomatic and/or untreated brain metastases (of any size or number) including active leptomeningeal or parenchymal metastases. Note: Participants with definitively treated brain metastases may be considered for enrollment if stable (defined as stable for 1-month post-central nervous system directed therapy) after discussion with the drug provider (KSQ) 9. Participants with ascites 10. Women who are pregnant or nursing 11. Seropositive for human immunodeficiency virus (HIV) 1 or 2 or acquired immunodeficiency syndrome, or active infection with hepatitis B virus (HBV) or hepatitis C virus (HCV) Note: Participants with positive HCV antibody may be eligible if HCV ribonucleic acid (RNA) is undetectable on a quantitative HCV RNA assay, following discussion with the drug provider (KSQ) 12. Any form of primary immunodeficiency (eg, Severe Combined Immunodeficiency Disease) 13. Any known clinically significant or concurrent acute liver disease, including viral hepatitis 14. Previous solid organ or hematopoietic cell transplant 15. Need for treatment with steroids at stable doses (\> 10 mg/day prednisone or equivalent) 1. Topical, ophthalmic, or inhaled steroid medications are allowed 2. Systemic steroid (\>10 mg/day) use is not allowed for 14 days prior to enrollment 3. Systemic steroids \< 10 mg/day are permitted if for supplemental endocrine only Note: steroids can be administered for IV contrast allergy 16. Live or unattenuated vaccine \< 28 days prior to first dose of LDC regimen 17. History of stroke, transient ischemic attack, unstable angina, or myocardial infarction, within 3 months prior to first dose of study treatment 18. Symptomatic congestive heart failure according to New York Heart Association (NYHA) classification, Class III or IV (per NYHA Classification), unstable angina pectoris, clinically significant cardia arrhythmia, or left ventricular ejection fraction \< 45% 19. Prolongation of QT/QTc interval (QTc interval \> 480 msec) using the Frederica method of QTc analysis 20. Unable to walk a distance of 80% predicted for age and sex or develop hypoxia (SPO2 \< 90%) during a 6-minute walk test (this test can be performed in place of pulmonary function test \[PFT\] for those unable to perform a reliable PFT due to complex upper airway anatomy) 21. For participants receiving IL-2 only: evidence of ischemia on cardiac stress test 22. \> 80% stenosis based on carotid doppler ultrasound for patients with NSCLC and HNSCC with \> 35 pack year smoking history 23. Obstructive or restrictive pulmonary disease Note: Post-bronchodilator values: forced expiratory volume (FEV1)/forced vital capacity \> 70% or FEV1 \> 50% of predicted normal are required for study entry 24. Suspected pneumonitis or interstitial lung disease (confirmed by radiography or computed tomography \[CT\]) 25. Treatment on another study with other investigational therapeutic interventional study within 28 days to start of LDC regimen 26. Known additional malignancy that is active and/or progressive requiring treatment; exceptions including basal cell or squamous cell skin cancer, or other cancer for which the participants has been disease-free for at least 2 years 27. Psychiatric illness/social situation that would limit compliance with study requirements, as determined by the Investigator 28. Other severe, acute, or chronic medical condition or laboratory abnormality that may increase the risk associated with study participation or study drug administration, or that may interfere with the interpretation of the study results, and in the judgement of the Investigator, would the participant inappropriate for the study