We isolate natural strengthenings of Bounded Martin’s Maximum which we call ${\mathsf{BMM}}^{*}$ and $A-{\mathsf{BMM}}^{*,++}$, and we investigate their consequences. We also show that if $A-{\mathsf{BMM}}^{*,++}$ holds true for every set of reals $A$ in $L$, then Woodin’s axiom $$ holds true. We conjecture that ${\mathsf{MM}}^{++}$ implies $A-{\mathsf{BMM}}^{*,++}$ for every $A$ which is universally Baire.