$\newcommand{\sym}{\mathfrak{S}}$ $\DeclareMathOperator{\Ext}{Ext}$
Let $k$ be a field of characteristic $p > 0$ and $a < p \leq b$ so that the $k\sym_a$-modules are semisimple but $k\sym_b$-modules are in general not.
Let $\lambda, \mu$ be partitions of $a$, and $\nu$ be a partition of $a+b$.
Let $S^\lambda, S_\mu$ be the corresponding Specht modules for $k\sym_a$.
Let $S_{\nu / \mu}$ be the dual skew Specht module for $k\sym_b$.
Writing $\boxtimes$ for the external tensor product, does $$\Ext^1_{\sym_a \times \sym_b}(S^\lambda \boxtimes k, S_\mu \boxtimes S_{\nu / \mu})$$ vanish? How about $\Ext^2$?
