I'm interested in forcing classes $\Gamma$ which preserve membership in themselves, i.e. for all posets $\mathbb{P}, \mathbb{Q}\in \Gamma$, we have $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}\in\Gamma$. For example, the class of countably closed forcing is self-preserving, since countably closed posets add no new $\omega$-sequences, so in particular they add no $\omega$-sequences without lower bounds to posets which were countably closed in the ground model.
In contrast, ZFC does not prove that ccc forcing is self-preserving, since forcing with a Suslin tree to add a cofinal branch makes it no longer have the countable chain condition. However, under $MA_{\omega_1}$ there are no Suslin trees, and in fact if $\mathbb{P}$ adds an antichain of size $\omega_1$ to $\mathbb{Q}$, then the existence of a filter meeting the dense sets of $\mathbb{P}$ which decide the members of that antichain implies that $\mathbb{Q}$ has an uncountable antichain in the ground model. Thus ccc forcing is self-preserving under $MA_{\omega_1}$.
On the other hand, forcing axioms are not always sufficient to guarantee self-preservation. Even under MM, Namba forcing and $Coll(\omega_1, \omega_2)$ are stationary set-preserving, but forcing with either one makes (the ground model version of) the other one collapse $\omega_1$.
In the case of proper forcing, the standard example seems to be that if there is a Suslin tree then both it and the forcing poset which specializes it are proper (in fact ccc), but each destroys the properness of the other. However, I'm having trouble figuring out what the situation is under PFA, where Suslin trees are not available.
Question: Is provable from or at least consistent with ZFC+PFA that there are proper posets $\mathbb{P}$ and $\mathbb{Q}$ such that $\Vdash_{\mathbb{P}}\check{\mathbb{Q}}$ is not proper?
