Study to Evaluate Effects of Vorinostat and HXTC on Persistent HIV-1 Infection in HIV-Infected Subjects Started on Antiretroviral Therapy (ART)

Inclusion Criteria: 1. ≥ 18 years and \< 65 years of age at screening 2. Ability and willingness of participant to give written informed consent. NOTE: Due to the lack of foreseeable benefit to study volunteers, the study will not enroll illiterate or mentally incompetent volunteers. 3. Confirmation of HIV-1 infection HIV infection is defined as documentation by any licensed rapid HIV test or HIV enzyme or chemiluminescence immunoassay (E/CIA) test kit at any time prior to study entry and confirmed by a licensed Western blot or a second antibody test by a method other than the initial rapid HIV and/or E/CIA, or by HIV-1 antigen, plasma HIV-1 RNA viral load. NOTE: The term "licensed" refers to a US FDA-approved kit. WHO (World Health Organization) and CDC (Centers for Disease Control and Prevention) guidelines mandate that confirmation of the initial test result must use a test that is different from the one used for the initial assessment. A reactive initial rapid test should be confirmed by either another type of rapid assay or an E/CIA that is based on a different antigen preparation and/or different test principle (e.g., indirect versus competitive), or a Western blot or a plasma HIV-1 RNA viral load. 4. On antiretroviral therapy for at least 24 months and on potent antiretroviral therapy for greater than or equal to 6 months prior to Screening (Visit 1). Potent ART is defined by current treatment guidelines and consists of at least 2 nucleoside/nucleotide reverse transcriptase inhibitors plus a non-nucleoside reverse transcriptase inhibitor, integrase inhibitor, or a protease inhibitor without interruption (defined as missing more than 9 total days in the 12 weeks prior to Screening. Other potent fully suppressive antiretroviral combinations will be considered on a case-by-case basis. Prior changes in or elimination of medications for easier dosing schedule, intolerance, toxicity, or other reasons are permitted if an alternative suppressive regimen was maintained. 5. Ability and willingness of participant to continue ART throughout the study. 6. Able and willing to adhere to protocol therapy, schedule, and judged adherent to antiretroviral therapy (adherence defined in inclusion criterion 4.) 7. Plasma HIV-1 RNA \< 50 copies/mL at two time points in the previous 12 months prior to study screening (one time point can be at screening) and never \> or equal to 50 copies/mL on two consecutive time points in the last 24 months. NOTE: A single unconfirmed plasma HIV RNA \> or equal to 50 copies/mL but \< 1000 c/mL is allowed if a subsequent assay was \< 50 copies/mL; but none in the 6 months preceding the study screening visit. 8. Plasma HIV-1 RNA \< 50 copies/mL at screening 9. CD4+ cell count ≥ 350 cells/mm3 at screening 10. No active HCV infection at or within 90 days of screening. Note: No active hepatitis C virus (HCV) defined as negative HCV antibody (HCVAb) or if HCVAb is positive, reflex HCV RNA is negative. 11. No active hepatitis B virus (HBV) infection (measureable HBV DNA or HBV surface antigen-positive (HBVsAg+)) at or within 90 days of screening 12. Women with written documentation of any of the following: 1. prior hysterectomy OR bilateral oophorectomy (removal of both ovaries) 2. bilateral tubal ligation or non-surgical permanent sterilization 3. Women with intact uterus and ovaries who have not had a period for ≥ one year AND have a documented follicle stimulating hormone (FSH) level indicating postmenopausal status. 13. All male study volunteers must agree not to participate in a conception process (e.g. active attempt to impregnate, sperm donation, in vitro fertilization) and, if participating in sexual activity that could lead to pregnancy, the male study volunteer and his female partner must use two reliable methods of contraception (condoms, with or without a spermicidal agent; a diaphragm or cervical cap with spermicide; an intrauterine device (IUD); or hormonal-based contraception) simultaneously while receiving the protocol-specified study products and for 6 weeks after stopping the study products. Participants must use a reliable barrier method of contraception (condom, cervical cap) along with another form of contraception. NOTE: For female partners who are receiving ritonavir, estrogen-based contraceptives are not reliable and an alternative method should be suggested. 14. Ability and willingness to provide adequate locator information. 15. Ability and willingness to communicate effectively with study personnel 16. Adequate vascular access for HXTC infusion and leukapheresis. 17. Able to swallow pills without difficulty. 18. Potential participant must have adequate organ function as indicated by the following laboratory values: System Laboratory Value Hematological Absolute neutrophil count (ANC) ≥1,500 /microliter (mcL) Platelets ≥125,000 / mcL Hemoglobin ≥ 12 g/dL (male) and ≥ 11.0 g/dL (females) System Laboratory Value Coagulation Prothrombin Time (PT) or international normalized ratio (INR) ≤1.1x upper limit of normal (ULN) Chemistry K+ levels within normal limits (WNL) Mg++ levels(footnote 1) WNL Glucose Screening serum glucose ≤ Grade 1 (fasting or non- fasting) Albumin ≥ 3.5 g/dL or ≥ lower limit of normal (LLN) Renal Creatinine clearance determined by the Chronic Kidney Disease Epidemiology Collaboration (CKD-Epi) equation found at: https://www.qxmd.com/calculate/calculator\_251/egfr-using-ckd-epi eGFR (epidermal growth factor receptor) \> 60 mL/min Hepatic Serum total bilirubin Total bilirubin \< 1.1 times the ULN range. If total bilirubin is elevated, direct bilirubin must be \< 2 times the ULN range. NOTE: If participant is on an atazanavir-containing therapy, then a direct bilirubin should be measured instead of the total bilirubin and must be ≤ 1.0 mg/dL. Aspartate Aminotransferase (AST) (SGOT) and Alanine Transaminase (ALT) (SGPT) \< 1.25 X ULN Alkaline Phosphatase \< 1.25 X ULN Lipase \< 1.1 X ULN Footnote 1: LLN for Mg++ per the clinical laboratory's normal range used for this study is a grade 1 event per Division of AIDS (DAIDS) Toxicity Table and is allowed for eligibility ULN = upper limit of normal LLN = lower limit of normal WNL = within normal limits Exclusion Criteria: 1. Known allergy or sensitivity to components of VOR and its analog or to components in the HXTC product. 2. Women without written documentation of menopause (absence of a period for ≥ one year and FSH level indicating menopause), hysterectomy or bilateral oophorectomy, non-surgical permanent sterilization, or bilateral tubal ligation. 3. Untreated syphilis infection (defined as a positive rapid plasma reagin (RPR) without clear documentation of treatment). Note: In cases of untreated syphilis, participant may re-screen following documentation of adequate treatment of syphilis 4. All male participants expecting to father children within the projected duration of the study. 5. Receipt of compounds with Histone Deacetylase (HDAC) inhibitor-like activity, such as valproic acid within 30 days prior to screening. 6. Use of any investigational antiretroviral agents within 30 days prior to screening. 7. If the study PI (or designee) is unable to construct a fully active alternative cART regimen based on previous resistance testing and/or treatment history. 8. Use of the following medications that carry risk of torsade des pointes: amiodarone, arsenic trioxide, astemizole, bepridil, chloroquine, chlorpromazine, cisapride, clarithromycin, disopyramide, dofetilide, domperidone, droperidol, erythromycin, halofantrine, haloperidol, ibutilide, levomethadyl, mesoridazine, methadone, pentamidine, pimozide, probucol, procainamide, quinidine, sotalol, sparfloxacin, terfenadine, thioridazine. 9. Use of any of the following within 90 days prior to screening: immunomodulatory, cytokine, or growth stimulating factors such as systemic corticosteroids, cyclosporine, methotrexate, azathioprine, anti-CD25 antibody, interferon (IFN), interleukin-2 (IL-2), coumadin, warfarin, or other Coumadin derivative anticoagulants. 10. Prior use of any HIV immunotherapy or HIV vaccine within 6 months prior to Screening, except for prior HXTC infusions. 11. Received any infusion blood product, immune globulin, or hematopoietic growth factors within 90 days prior to study screening. 12. Pregnancy or breast-feeding. 13. History or other clinical evidence of severe illness, malignancy, immunodeficiency other than HIV, or any other condition that would make the participant unsuitable for the study in the opinion of the investigator (or designee). 14. Use of topical steroids over a total area exceeding 15 cm-2 within 30 days prior to Screening. 15. Treatment for an active AIDS-defining opportunistic infection within 90 days prior to Screening. 16. Any active malignancy that may require chemotherapy or radiation therapy. 17. Compulsorily detained (involuntarily incarcerated) for treatment of either a psychiatric illness or a physical illness, e.g., infectious disease. Prisoner recruitment and participation is not permitted. 18. Known psychiatric or substance abuse disorders that would interfere with participant's ability to fully cooperate with the requirements of the trial as assessed by the study investigator (or designee). 19. History or other clinical evidence, as assessed by the study PI (or designee), of significant or unstable cardiac disease (e.g., angina, congestive heart failure, recent myocardial infarction, significant arrhythmia requiring medical or surgical therapy) or clinically significant electrocardiogram (ECG) abnormalities. 20. Unable to have a person available to drive participant home at infusion visits. 21. Participation in another investigational clinical research study (with the exception of an antiretroviral treatment trial that uses FDA approved antiretroviral agents) or use of investigational agents within 30 days prior to screening. NOTE: Co-enrollment in observational only studies is permitted. NOTE: Co-enrollment in the AIDS Clinical Trials Group (ACTG) 5332 REPRIEVE study (NCT023442900) and using FDA approved pitavastatin is permitted provided participant enrolled on ACTG 5332 and has taken the study provided medication ≥ 4 months.