The Tree Property at $\omega _2$ and Bounded Forcing Axioms

We prove that the Tree Property at $\omega _2$ together with $\mathrm {BPFA}$ is equiconsistent with the existence of a weakly compact reflecting cardinal, and if $\mathrm {BPFA}$ is replaced by $\mathrm {BPFA}(\omega _1)$ then it is equiconsistent with the existence of just a weakly compact cardinal. Similarly, we show that the Special Tree Property for $\omega _2$ together with $\mathrm {BPFA}$ is equiconsistent with the existence of a reflecting Mahlo cardinal, and if $\mathrm {BPFA}$ is replaced by $\mathrm {BPFA}(\omega _1)$ then it is equiconsistent with the existence of just a Mahlo cardinal.