Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}_\alpha$ be the countable support iteration of $\mathbb{P}$ of length $\alpha$.

It is well known that $\mathbb{P}_\alpha$ is also proper. Hence for any countable elementary structure $M \prec H_\Theta$ for sufficiently large $\Theta$ such that $\mathbb{P}_\alpha \in M$ and any $p \in \mathbb{P}_\alpha \cap M$, there exists some $q \leq_{\mathbb{P}_\alpha} p$ which is a $(M, \mathbb{P}_\alpha)$-generic condition.

Let $X = \alpha \cap M$. The question is whether a $(M, \mathbb{P}_\alpha)$ generic condition $q \leq_{\mathbb{P}_\alpha} p$ can be found such that $\text{supp}(q) \subseteq X$.

Thanks for any information that can be provided.