Forcing "${\mathrm{N}\mathrm{S}}_{{\omega }_1}$ is ${\omega }_1$-dense" From Large Cardinals

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of $<\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb{P}}\models \mathrm{‘}\mathrm{‘}{\mathrm{N}\mathrm{S}}_{{\omega }_1}\text{ is }{\omega }_1\text{-dense}\mathrm{"}$. We also introduce a new forcing axiom $\mathrm{Q}\mathrm{M}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies ${\mathbb{Q}}_{\mathrm{m}\mathrm{a}\mathrm{x}}\text{-}(\ast )$. Consequently, $\mathrm{Q}\mathrm{M}$ implies ``${\mathrm{N}\mathrm{S}}_{{\omega }_1}$ is ${\omega }_1$-dense".