Preserving distributivity with finite support products We work in $\sf ZFC+GCH$. Let $D$ be a class of uncountable regular cardinals, and for every $\alpha$ let $\Bbb Q_\alpha$ be either trivial if $\alpha\notin D$, or forcing with these two properties:


*

*$|\Bbb Q_\alpha|=\alpha$, and

*$\Bbb Q_\alpha$ is $\alpha$-distributive (any less than $\alpha$ dense open sets have a dense intersection).


Let $\Bbb P_\alpha$ be the finite support product of $\Bbb Q_\beta$ for $\beta<\alpha$. 

Suppose that $\alpha\in D$, is there are condition which implies that $\Bbb P_\alpha$ preserves the truth value of "$\Bbb Q_\alpha$ does not add bounded sets to $\alpha$"?

The condition can be made on either the forcings themselves, or on $D$ (e.g. $(\sup D\cap\alpha)^+<\alpha$ or something like that).
 A: GCH implies that $\mathbb{Q}_\alpha$ is $\alpha$-distributive in the generic extension by $\mathbb{P}_\alpha$. 
Note that $\Vdash_{\mathbb{Q}_\alpha} ``\check{\mathbb{P}}_\alpha$ is $\check\alpha$.c.c.$"$. For successor cardinal $\alpha$ this is clear, as $|\mathbb{P}_\alpha| < \alpha$. For strongly inaccessible $\alpha$, the strong inaccessibility of $\alpha$ is preserved under $\mathbb{Q}_\alpha$. Therefore, was can apply the usual $\Delta$-system argument in the generic extension by $\mathbb{Q}_\alpha$ and conclude that $\mathbb{P}_\alpha$ is still $\alpha$-c.c. (and even $\alpha$-Knaster).
Let $\dot{\tau}$ be a $\mathbb{P}_\alpha \times \mathbb{Q}_\alpha$-name for a new set of ordinals of size $\mu$ for some $\mu <\alpha$. Let $G\subseteq \mathbb{Q}_\alpha$ be a generic filter. In $V[G]$, $\mathbb{P}_\alpha$ is $\alpha$.c.c. and $\alpha$ is still regular. Therefore, there is a $\mathbb{P}_\alpha$-name $\dot{\sigma}\in V[G]$, $$V[G] \models |\dot{\sigma}| = \nu < \alpha,\ \text{and }1_{\mathbb{P}_\alpha}\Vdash_{\mathbb{P}_\alpha} \dot{\tau} = \dot{\sigma}.$$ 
But $\mathbb{Q}_\alpha$ is $\alpha$-distributive, and therefore $\dot{\sigma}\in V$. We conclude that there is a condition $q\in \mathbb{Q}_\alpha$ such that $(1_{\mathbb{P}_\alpha}, q) \Vdash \dot{\tau} = \dot{\sigma}$ and in particular $(1_{\mathbb{P}_\alpha}, q)\Vdash \dot{\tau}\in V^{\mathbb{P}_\alpha}$.
