Stable character bounds for word measures on wreath products

Let $G$ be a finite group, let $F_r$ be the free group of rank $r$, and let $w\in F_r$. For a stable irreducible character $\chi=(\chi_n)_{n\in\mathbb{N}}$ of $G\wr S_{\bullet}$, write $\dim(\chi)=(\dim(\chi_n))_{n\in\mathbb{N}}$ and $\operatorname{\mathbb{E}}_w[\chi]=(\operatorname{\mathbb{E}}_w[\chi_n])_{n\in\mathbb{N}}$. Stable character bound. For every stable irreducible character $\chi$ and $w\in F_r$,

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

where $n\to\infty$ and the implied constant depends on $\chi,w$. This conjecture predicts that word-measure expectations decay according to the primitivity rank, extending known character estimates for symmetric and wreath products; its resolution is not specified here.