Martin's Maximum${}_{\mathfrak{c}}^{\ast ,++}$ in ${\mathbb{P}}_{\max }$ extensions of strong models of determinacy

We study a strengthening of ${\mathrm{M}\mathrm{M}}^{++}$ which is called ${\mathrm{M}\mathrm{M}}^{\ast ,++}$ and which was introduced by Asperó and Schindler. We force its bounded version ${\mathrm{M}\mathrm{M}}_{\mathfrak{c}}^{\ast ,++}$, which is stronger than both ${\mathrm{M}\mathrm{M}}^{++}(\mathfrak{c})$ as well as ${\mathrm{B}\mathrm{M}\mathrm{M}}^{++}$, by ${\mathbb{P}}_{\max }$ forcing over a determinacy model $L^{F_{\mathrm{u}\mathrm{B}}}({\mathbb{R}}^{\ast },Ho{m}^{\ast })$. The construction of the ground model $L^{F_{\mathrm{u}\mathrm{B}}}({\mathbb{R}}^{\ast },Ho{m}^{\ast })$ builds upon Gappo and Sargsyan, and the derived model construction of Larson, Sargsyan, and Wilson.