Amit–Vishne–Shalev conjecture on profinite rigidity of free-group words

Let $F_n=\langle x_1,\ldots,x_n\rangle$ be the free group on $n$ generators. For a word $w\in F_n$, let $\operatorname{Pr}_w(g)$ be the probability that $w(g_1,\ldots,g_n)=g$ when $(g_1,\ldots,g_n)$ is chosen uniformly from a finite group $G^n$. Call $w$ profinetely rigid if every word $u\in F_n$ inducing the same probability measure as $w$ on every finite group is automorphic to $w$. Amit–Vishne–Shalev conjecture. Every word in $F_n$, for $n\geq 2$, is profinitely rigid. The conjecture asserts that the induced probability measures of a word on all finite groups determine its orbit under automorphisms of the free group. It remains open in general, although recent developments support it.