A Phase 1/2a Multicenter Ascending Dose Study to Evaluate the Safety of HA-1 Minor Histocompatibility Antigen-Reactive TCR-Modified T Cells (BSB-1001) in Patients Undergoing HLA-Matched Allogenic Hematopoietic Stem Cell Transplant for AML, ALL or MDS

Inclusion Criteria: 1. Male or female patients, ages 18 - 70 years inclusive, undergoing alloHCT. 2. Any of the following high-risk hematologic malignancies: 1. AML diagnosed which has been treated with at least two lines of therapy\* Refractory or relapsed (CR, CRh or CRi,), including myeloblasts up to 25% OR MRD positive OR persistent disease-defining cytogenetic abnormality OR MRD-negative, but with high-risk disease For patients in remission meeting criteria a, consolidation regimens would be considered another line of therapy of eligibility purposes 2. ALL which has been with abnormal lymphoblasts ≥5% and up to 25% in bone marrow OR persistent disease-defining cytogenetic abnormality or MRD positive 3. MDS after at least one line of therapy, which includes hypomethylating agent(s) and must be high or very high risk by Revised International Prognostic Scoring System (IPSS-R), monosomy, or complex karyotype or TP53 mutation. 4. In the expansion phase AML patients diagnosed which has been treated with at least two lines of therapy, and refractory or relapsed (CR, CRh or CRi,), including myeloblasts up to 25% OR MRD positive OR persistent disease-defining cytogenetic abnormality OR MRD-negative, but with high- risk disease 3. HLA-A\*02:01 AND HA-1 positive (either H/H or H/R). 4. Suitable for one of the approved conditioning regimens as defined in the protocol. 5. Patient must have an identified donor that is HA 1-negative with 10/10 matched related or unrelated donor Exclusion Criteria: 1. Weight \> 100 kg. 2. Prior history of allogeneic stem cell transplantation 3. Prior history of autologous stem cell transplantation within 1 year prior to the planned dosing of BSB-1001 (day 0) 4. Previous genetically engineered chimeric antigen receptor T Cell therapy (CAR-T), approved or investigational, within 2 years of screening, with the exception of patients with ALL previously treated with an autologous CAR-T product. 5. Treatment with other investigational agents within 5 half-lives of the planned dosing of BSB-1001 (day 0). 6. History of treatment with checkpoint inhibitor therapy within 3 months of transplantation. 7. Other malignancy with life expectancy \< 1year. 8. Pregnant or lactating women. 9. Uncontrolled bacterial, viral, or fungal infections at time of enrollment. 10. Past or current viral infections as defined in the protocol. 11. CNS involvement refractory to intrathecal chemotherapy and/or standard cranial- spinal radiation. 12 Karnofsky Performance Score \< 60%. 13\. Inadequate organ function as defined in protocol.