Bounded Martin’s Maximum with an Asterisk

We isolate natural strengthenings of Bounded Martin’s Maximum which we call BMM∗BMM∗ and A−BMM∗,++A−BMM∗,++ (where AA is a universally Baire set of reals), and we investigate their consequences. We also show that if A−BMM∗,++A−BMM∗,++ holds true for every set of reals AA in L(R)L(R), then Woodin’s axiom (∗)(∗) holds true. We conjecture that MM++MM++ implies A−BMM∗,++A−BMM∗,++ for every AA which is universally Baire.