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$\newcommand{\sym}{\mathfrak{S}}$ $\DeclareMathOperator{\Ext}{Ext}$

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 $\Ext^2$?

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