Stable primitivity-rank formula for symmetric-group character averages

Let $S_{\bullet}=\{S_N\}_{N\ge 1}$, and let $\chi^{\mu[\bullet]}=\{\chi^{\mu[N]}\}$ denote the stable irreducible character associated with a partition $\mu$. For a word $w$ in the free group $\mathbb{F}$, let $\beta(w,\chi^{\mu[\bullet]})$ be defined by the asymptotic relation

$$\mathbb{E}_{w}\left[\chi^{\mu[N]}\right]=\left(\dim\chi^{\mu[N]}\right)^{-\beta(w,\chi^{\mu[\bullet]})}\left(c+O\left(N^{-1}\right)\right)$$

for some constant $c\ne0$, and let $\mathrm{s\pi}(w)$ be the stable primitivity rank of $w$. Let ${\cal I}=\left\{\chi^{\mu[\bullet]}\right\}_{\mu}$ be the set of stable irreducible characters of $S_{\bullet}$. Stable primitivity-rank conjecture. For every $w\in\mathbb{F}$,

$$\inf_{\mathrm{triv}\ne\chi\in{\cal I}}\beta(w,\chi)=\mathrm{s\pi}(w).$$

Moreover, the infimum on the left-hand side is attained. This would identify stable primitivity rank as the optimal decay exponent among non-trivial stable irreducible character averages for symmetric groups; the supplied text gives no resolution status.