Hiss–Schröer bounded multiplicity conjecture for finite classical groups

Let $G={\mathrm{Sp}}_{2n}$, ${\mathrm{U}}_n$, or ${\mathrm{GL}}_n$ be defined over a finite field $\mathfrak{f}$, let $q=|\mathfrak{f}|$, and let $\pi\in\operatorname{Irr}(G^F)$. Write

$$\pi\otimes\omega_\psi^\vee=\sum_{\sigma\in\operatorname{Irr}(G^F)}m(\pi,\sigma)\sigma.$$

Hiss–Schröer's conjecture. The multiplicities $m(\pi,\sigma)$ of the irreducible constituents of $\pi\otimes\omega_\psi^\vee$ are bounded by a function of $n$ independent of $q$.

This conjecture asks for uniform multiplicity bounds in the basic Fourier–Jacobi model as the size of the defining finite field varies. The source presents it as one of two general conjectures of Hiss and Schröer; no resolution is stated here.