  We study a strengthening of $\mathrm{MM}^{++}$ which is called
$\mathrm{MM}^{\ast, ++}$ and which was introduced by Asper\'o and Schindler. We
force its bounded version $\mathrm{MM}^{\ast, ++}_{\mathfrak{c}}$, which is
stronger than both $\mathrm{MM}^{++}(\mathfrak{c})$ as well as
$\mathrm{BMM}^{++}$, by $\mathbb{P}_{\max}$ forcing over a determinacy model
$L^{F_{\mathrm{uB}}}({\mathbb R}^*,\mbox{Hom}^{\ast})$. The construction of the
ground model $L^{F_{\mathrm{uB}}}({\mathbb R}^{\ast},\mbox{Hom}^{\ast})$ builds
upon Gappo and Sargsyan, and the derived model construction of Larson,
Sargsyan, and Wilson.
