Larsen–Shalev conjecture for automorphic word maps

Let $w$ be a nonempty reduced word in $d$ distinct variables, let $S$ be a nonabelian finite simple group, and let $\mathfrak{P}_w(S)$ denote the relevant automorphic word-map quantity.

Larsen–Shalev conjecture. For each such word $w$, there exist constants $N(w),\eta(w)>0$ such that, for every nonabelian finite simple group $S$ with $|S|\geq N(w)$,

$$\mathfrak{P}_w(S)\leq |S|^{d-\eta(w)}.$$

This is presented as the direct automorphic-word-map analogue of Larsen and Shalev's result for ordinary word maps. The statement is intended to follow from an adaptation of their proof, but the bounded-rank Lie-type case remains open because field automorphisms can make the defining degrees of fibers unbounded.