Equidistribution from large image for word maps on Chevalley groups

For a fixed prime $p$, let $G_q$ be a family of Chevalley groups of fixed Lie type over $\mathbb{F}_q$, where $q=p^n$ varies. For a fixed word $w\in F_d$, with $d\ge 2$, let

$$P_q=P_{w,q}\colon (G_q)^d\to G_q$$

be the corresponding word map. Assume that, for all sufficiently large $n$, the image of $P_q$ contains every regular semisimple element of $G_q$. Large-image equidistribution conjecture. Then the family $\{P_q\}$ is almost $p$-equidistributed. This would extend the paper’s comparison between large image and equidistribution from two-letter words and the groups considered there to families of Chevalley groups of arbitrary fixed Lie type; the claim remains open in the source.