Stable-character expectation bound for word measures on general linear groups

Let $K$ be a finite field with $|K|=q$, let $\mathbb{F}$ be a free group, and let $w\in\mathbb{F}$. Let $\mathcal{R}$ be the algebra of character-like functions on the groups $\operatorname{GL}_N(K)$, and let $\chi$ be a stable character of $\operatorname{GL}_\bullet(K)$, meaning an element of $\mathcal{R}$ that coincides, for every sufficiently large $N$, with an irreducible character of $\operatorname{GL}_N(K)$. Stable-character expectation conjecture.

$$\mathbb{E}_w[\chi]=O\left((\dim\chi)^{1-\pi_q(w)}\right).$$

This extends the fixed-vector conjecture from the natural representation to stable irreducible character families. The source places the claim in the framework of stable representations of $\operatorname{GL}_\bullet(K)$; the general bound is not established there.