  We study a strengthening of Bounded Martin's Maximum which asserts that if a
\Sigma_1 fact holds of \omega_2^V in a stationary set preserving extension then
it holds in V for a stationary set of ordinals less than \omega_2. We show that
this principle implies Global Projective Determinacy, and therefore does not
hold in the \mathbb{P}_{max} model for \mathsf{BMM}, but that the restriction
of this principle to forcings which render \omega_2^V countably cofinal does
hold in the \mathsf{BMM} model, though it is not a consequence of \mathsf{BMM}.
