The alternating-group bismash-product Schur indicator conjecture

For $n\geq 3$, let $H=\Bbbk^{C_2}\#\Bbbk A_n$, and let $\nu^{[2]}(\widehat{\chi})$ denote the Schur indicator of an irreducible character $\widehat{\chi}$ of $H$.

Alternating-group indicator conjecture. The indicator satisfies

$$\nu^{[2]}(\widehat{\chi})\in\{0,1\}.$$

The analogous statement for $\Bbbk^{A_n}\#\Bbbk C_2$ is proved for $n\geq4$. The dual statement is motivated by twisted Frobenius–Schur indicator theory and had been checked computationally through $n=13$, but the general case remains open.