This phase III trial studies rituximab after stem cell transplant and to see how well it works compared with rituximab alone in treating patients with in minimal residual disease-negative mantle cell lymphoma in first complete remission. Immunotherapy with rituximab, may induce changes in bodys immune system and may interfere with the ability of tumor cells to grow and spread. Giving chemotherapy before a stem cell transplant helps kill any cancer cells that are in the body and helps make room in the patients bone marrow for new blood-forming cells (stem cells) to grow. After treatment, stem cells are collected from the patient's blood and stored. More chemotherapy is then given to prepare the bone marrow for the stem cell transplant. The stem cells are then returned to the patient to replace the blood-forming cells that were destroyed by the chemotherapy. Giving rituximab with or without stem cell transplant may work better in treating patients with mantle cell lymphoma.

This study will evaluate the safety, tolerability, and efficacy of an engineered donor graft
("Orca-T", a T-cell-Depleted Graft With Additional Infusion of Conventional T Cells and
Regulatory T Cells) in participants undergoing myeloablative allogeneic hematopoietic cell
transplant transplantation for hematologic malignancies.