The exceptional symmetric-group factorization conjecture for bismash products

Let $S_n=FG$ be an exact factorization of $S_n$ with $F\triangleq\widetilde{S}_{n-k}$ as in the classified cases for $k=2$ or $3$, and let $G$ be sharply $k$-transitive. Set

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

The exceptional factorization conjecture. The bismash product $H$ has an irreducible module with Schur indicator $-1$.

The claim predicts that the exceptional complements of the symmetric groups yield the negative Schur indicators that do not occur for the ordinary symmetric-group factorizations. The displayed example establishes this phenomenon for one factorization, while the stated general case remains conjectural.