Universal word-measure character bound for natural families of groups

Let $G=\{G(N)\}_N$ be a natural family of groups, and let $\chi=\{\chi_N\}_{N\geq N_0}$ be a stable irreducible character with $\chi_N\in\widehat{G(N)}$. Let $\pi(w)$ denote the primitivity rank of a word $w\in\mathbb{F}$, and let $\dim\chi$ denote the dimension of the corresponding representation. Universal word-measure character conjecture. For any word $w\in\mathbb{F}$, as $N\to\infty$,

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

Here the implied constant may depend on $w$, on $G$, and on $\chi$. This generalizes the analogous conjecture for stable irreducible characters of symmetric groups to other natural families, including unitary, orthogonal, compact symplectic, finite general linear, and generalized symmetric groups. The source gives no resolution.