The integrality and self-duality conjecture for Clifford-group characters

Let $\mathcal{C}_n$ be the $n$-qubit Clifford group, and let its character table consist of the values of its irreducible characters on its conjugacy classes.

Integrality and self-duality conjecture. The entries in the character table of $\mathcal{C}_n$ are integers, and hence all irreducible characters of $\mathcal{C}_n$ are self-dual.

This is verified for many conjugacy classes, including those containing $H_1$, $P_1$, $P_1^2$, $H_1P_1$, and $Z_1$. The asserted property would further imply that all irreducible representations of $\mathcal{C}_n$ are self-dual, paralleling a notable property of symmetric groups.