Let $FA_{\aleph_1}$(cardinal preserving proper forcings) be the forcing axiom: if $\mathbb{P}$ is a cardinal preserving proper forcing notion and $(D_\xi)_{\xi<\omega_1}$ are dense subsets of $\mathbb{P},$ then there exists a filter $G \subseteq \mathbb{P}$ which meets all $D_\xi$'s, $\xi < \omega_1$.

Does this forcing axiom decide CH. More precisely,

Question. Assuming the existence of large cardinals, is $FA_{\aleph_1}$(cardinal preserving proper forcings) consistent with large values of the continuum?