Let $\mathsf{M}$ be a simplicial model category presenting an $\infty$-category $\mathcal{M}$. I'm interested in a general statement relating compact objects in $\mathcal{M}$ (in the $\infty$-categorical sense) with the compact objects in $\mathsf{M}$. Here's roughly what I expect to be true but if i'm missing some sumptions or some are redundant feel free to phrase the correct statement as an answer.

Suppose further that $\mathsf{M}$ satisfies that weak equivalences between fibrant objects are stable under filtered colimits. Then is the following true

1. Let $X$ be a compact cofibrant object in $\mathsf{M}$. Is $X$ compact as an object in $\mathcal{M}$?

2. Is every compact object in $\mathcal{M}$ a retract of (the image in $\mathcal{M}$) of some compact object in $\mathsf{M}$?