The word-measure rigidity conjecture for free groups

Let $\mathbf{F}_r$ be the free group of rank $r$, and let $w\in\mathbf{F}_r$ induce a probability measure on each compact group via its word map. Word-measure rigidity conjecture. If two words $w_{1},w_{2}\in\mathbf{F}_r$ induce the same measure on every compact group, then there exists $\phi\in\operatorname{Aut}(\mathbf{F}_r)$ with $w_{2}=\phi(w_{1})$. This asks whether equality of word measures on all compact groups can occur only for words related by an automorphism of the free group; it was settled affirmatively by Puder and Parzanchevski using word measures on symmetric groups.