The non-exceptional symmetric-group factorization conjecture for Schur indicators

Let $S_n=F\cdot G$ be an exact factorization with $|F|>|G|$, and suppose $F$ is not one of the exceptional groups $\widetilde{S}_{n-k}$ in the classified cases. Set

$$H=\Bbbk^G\#\Bbbk F.$$

Non-exceptional factorization conjecture. The Schur indicator of every irreducible $H$-module is $0$ or $1$.

This would identify the exceptional factorizations as the only badly behaved factorizations. The assertion is supported by the theorem covering factorizations with $n\leq10$ under the additional stated cycle hypothesis, but is open in general.