HIV Antigen-specific T-cells Targeting Conserved Epitopes (HST-NEETs) BMTCTN1903

Eligible participants will have 100-120 mL of peripheral blood or 80-100 mL of MNCs via apheresis collected and shipped to Children's National Hospital at ambient temperature. The sample will be used to manufacture the HST-NEET product. The autologous peripheral blood stem cell graft suitable for rescue following conditioning will be obtained either before or after the collection of blood to generate HST-NEETs. Pre-transplant conditioning will consist of BEAM; BCNU 300 mg/m\^2 on Day -6, Etoposide 100 mg/m\^2 BID and Ara-C 100 mg/m2 BID on Days -5, -4, -3 and -2 and Melphalan 140 mg/m2 on Day -1. ASCT on Day 0. If the mobilized graft contains greater than 5.0 x 106 CD34+ cells per kg, any additional cells should be cryopreserved as a "back-up" graft in the event of graft failure related to the HST-NEETs. Participants will receive one dose (2 x 107 cells/m\^2 ) of HST-NEETs between Days +3 to +7 based on the clinical condition of the participant. If this window is missed, the HST-NEETs may be administered up to Day +30 post-ASCT. Participants will be followed for at least one year after ASCT.