Let $T$ be a first-order theory with two designated unary predicates $P$ and $Q$. We'll say that $P$ is bounded over $Q$ if for every $\kappa$, there is a $\lambda$ such that if $M \models T$ and $|Q^M| \leq \kappa$, then $|P^M|\leq \lambda$. Vaught's theorem on cardinals far apart tells us that $P$ is bounded over $Q$ if and only if there is a $k < \omega$ such that $|P^M| \leq \beth_k(|Q^M|+|T|)$ for all $M\models T$.
We'll say that $P$ is strictly bounded over $Q$ if for every $M \models T$, $|P^M| \leq |Q^M|$. Vaught's two-cardinal theorem tells us that strict boundedness is a significantly stronger condition than boundedness. In particular, if $P$ is strictly bounded over $Q$, then for any pair of models $M,N\models T$ with $M \preceq N$, if $Q^M=Q^N$, then $P^M = P^N$.
The basic example of a theory in which $P$ is bounded over $Q$ but not strictly bounded is the two-sorted structure $M=(X,2^X,\in)$ with $X$ an infinite set. If we take $P^M = 2^X$ and $Q^M = X$, then we clearly have that $P$ cannot be arbitrarily large relative to $Q$ in elementary extensions of $M$.
There is a relevant result of Shelah, which states that in a stable theory, if $P$ is bounded over $Q$, then it is strictly bounded. My question has to do with the extent to which this assumption can be weakened.
Recall that $Q$ is stably embedded if any definable subset of $Q^n$ is definable with parameters from $Q$. All of the examples (that I can think of) of theories in which $P$ is bounded but not strictly bounded over $Q$ fail to have $Q$ be stably embedded, so it seems reasonable to me to ask whether this is actually necessary.
Question. If $P$ is bounded over $Q$ and $Q$ is stably embedded, is it necessarily true that $P$ is strictly bounded over $Q$?
