Hanany–Puder character bound conjecture for word maps on symmetric groups

Let $F_m$ be the free group of rank $m$. For a word $w\in F_m$, let

$$\pi(w)=\min\{\operatorname{rk}H:H\leq F_m,\ w\in H,\text{ and }w\text{ is not primitive in }H\},$$

with $\pi(w)=\infty$ if no such subgroup exists. For $k\in\mathbb{Z}_{\geq 2}$ and a partition $\lambda\vdash k$, let $\lambda^{+}(n)$ denote the corresponding stable Young diagram, and let $\chi^{\lambda^{+}(n)}$ and $V^{\lambda^{+}(n)}$ be the associated irreducible character and representation of $S_n$. For independent uniformly random permutations $\sigma_1,\dots,\sigma_m\in S_n$, write $w(\sigma_1,\dots,\sigma_m)$ for the word-map evaluation.

Hanany–Puder conjecture. For any $k\in\mathbb{Z}_{\geq2}$, any $\lambda\vdash k$, and any $w\in F_m$,

$$\mathop{\mathbb{E}}_{\sigma_{1},\dots,\sigma_{m}\in S_{n}}\left[\chi^{\lambda^{+}(n)}\left(w\left(\sigma_{1},\dots,\sigma_{m}\right)\right)\right]=O\left(\frac{1}{\left(\dim V^{\lambda^{+}(n)}\right)^{\pi(w)-1}}\right).$$

This conjectures a substantially stronger decay estimate than the known bound $O(n^{-\pi(w)})$ for words that are not proper powers or the identity. It refines the study of distributions of word maps on symmetric groups by relating character expectations to the primitivity rank of the word and the dimension of the corresponding stable irreducible representation.