Let $\kappa$ be strongly inaccessible, and let $\mu<\kappa$ be regular. What is the effect on $\kappa$ of forcing with the following?
(1) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, with supports of size $<\kappa$. (bounded support)
(2) The product of $Col(\mu,\alpha)$ for $\alpha<\kappa$, with supports of size $\leq \kappa$. (full support)
The failure of $\kappa$-cc is clear, but do these collapse $\kappa$?