Consistency of LSA from the consistency of T and PFA

Let $\textsf{LSA}$ denote the theory “$\textsf{AD}^+ + \theta_{\beta+1}+\theta_\beta$ is the largest Suslin cardinal.” Here $\operatorname{Con}(T)$ and $\operatorname{Con}(\textsf{PFA})$ express the consistency of $T$ and $\textsf{PFA}$, respectively. Consistency-strength conjecture.

$$\operatorname{Con}(T)\Rightarrow\operatorname{Con}(\textsf{LSA})\quad\text{and}\quad\operatorname{Con}(\textsf{PFA})\Rightarrow\operatorname{Con}(\textsf{LSA}).$$

The conjecture proposes that both $T$ and $\textsf{PFA}$ have at least the consistency strength of $\textsf{LSA}$, a strong determinacy theory whose consistency was recently established according to the source. The relative consistency implications remain conjectural.