Hiss–Schröer conjecture on bounded Fourier–Jacobi multiplicities

Let $q$ be a power of a prime $p\neq 2$, let $\psi:\mathbb F_q\to\mathbb{\bar Q}_\ell^\times$ be a nontrivial additive character, and let $G_n=\mathrm U_n$ or $\mathrm{Sp}_{2n}$. For irreducible representations $\pi,\pi'$ of the finite group $G_n(\mathbb F_q)$, write $\omega_{G_n(\mathbb F_q),\psi}$ for the Weil representation corresponding to $\psi$. Hiss and Schröer's conjecture. There is a function $f:\mathbb Z_+\to\mathbb Z_+$ independent of $q$ such that

$$\langle \pi,\pi'\otimes\omega_{G_n(\mathbb F_q),\psi}\rangle_{G_n(\mathbb F_q)}\leq f(n)$$

for all such $\pi$ and $\pi'$. The paper states that its results prove this conjecture, yielding a bound on Fourier–Jacobi multiplicities independent of $q$; hence the conjecture is solved.