Nonvanishing signature conjecture for quantum representations

Let $p$ be a prime with $p\geq 5$, let $\zeta$ be a primitive $2p$-th root of unity, and define

$$A_p=(-1)^{\frac{p-1}{2}}\exp\left(\frac{(p+1)\pi i}{2p}\right)$$

for odd $p$, and $A_p=\pm\exp\left(\frac{\pi i}{p}\right)$ for even $p$. Assume that neither $\zeta$ nor $\overline{\zeta}$ equals $A_p$. Nonvanishing signature conjecture. For all $g$ in some arithmetic progression, the signature $\sigma(g,\zeta)$ of the corresponding Hermitian form is nonzero modulo $p$:

$$\sigma(g,\zeta)\not\equiv 0 \pmod p.$$

The claim is presented as evidence-based motivation for further study of signatures of quantum representations; the source does not provide a resolution, so its status remains open.