Androgen Deprivation Therapy (Relugolix) for the Improvement of Diagnostic Imaging (PSMA PET/CT Scan) in Patients With High Risk or Very High Risk Prostate Cancer, The EnrichPSMA Trial

Inclusion Criteria: * Age ≥ 18 years * Histological confirmation of prostate adenocarcinoma * Diagnosis of high risk or very high risk prostate cancer per National Comprehensive Cancer Network (NCCN) Risk Stratification. \[Any of the following: grade group 4 or 5, prostate-specific antigen (PSA) greater than 20, radiographic cT3 on MRI\] * Testosterone greater than or equal to 300 * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2 * Hemoglobin ≥ 9.0 g/dL (obtained ≤ 120 days prior to registration/randomization) * Absolute neutrophil count (ANC) ≥ 1500/mm\^3 (obtained ≤ 120 days prior to registration/randomization) * Platelet count ≥ 100,000/mm\^3 (obtained ≤ 120 days prior to registration/randomization) * Male patients who are committed to undertaking the following measures for the duration of the study and after the last dose of ORGOVYX (relugolix) for the time period specified: * Use a condom during sex while being treated and for 30 days after the last dose of ORGOVYX (relugolix) * Do not make semen donations during treatment and for 30 days after the last dose of ORGOVYX (relugolix) * Those with female partners of childbearing potential may be enrolled if they are: * Documented to be surgically sterile (i.e., vasectomy); * Committed to practicing true abstinence during treatment and for 30 days after the last ORGOVYX (relugolix) dose; or * Committed to using an effective method of contraception with their partner during treatment and for 30 days following the last dose of ORGOVYX (relugolix) * Provide written informed consent Exclusion Criteria: * Any of the following prior therapies: * Chemotherapy ≤ 2 weeks prior to registration/randomization * Androgen deprivation therapy * Pelvic radiation * Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens * Uncontrolled intercurrent illness including, but not limited to: * Ongoing or active infection * Symptomatic congestive heart failure * Unstable angina pectoris * Cardiac arrhythmia * Or psychiatric illness/social situations that would limit compliance with study requirements * Receiving any other investigational agent which would be considered as a treatment for the primary neoplasm * Other active malignancy ≤ 1 year prior to registration * EXCEPTIONS: Non-melanotic skin cancer * NOTE: If there is a history of prior malignancy, they must not be receiving other active treatment for their cancer * History of myocardial infarction ≤ 6 months, or congestive heart failure requiring use of ongoing maintenance therapy for life-threatening ventricular arrhythmias * Use of P-glycoprotein inhibitors