It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ where $\kappa$ is a $\Sigma_2$-correct cardinal.

My questions are about what large cardinal principles can prove $\Sigma_3^1$-generic absoluteness. In particular:

1) If $0^\sharp$ exists (or even $x^\sharp$ exists for all reals $x$), does (light-face) $\Sigma_3^1$-generic absoluteness holds.

2) What if there is a measurable cardinal, then does $\Sigma_3^1$-generic absoluteness hold?

Is there any large cardinal whose existence implies $\Sigma_3^1$ generic-absoluteness?