The word-measure orbit conjecture

Let $F_r$ be the free group on $r$ generators. For $w,w'\in F_r$ and a compact group $G$, let $\mu_{w,G}$ be the push-forward of Haar measure on $G^r$ under the word map determined by $w$. The measure $\mu_{w,G}$ depends only on the $\operatorname{Aut}(F_r)$-orbit of $w$. Word-measure orbit conjecture. If

$$\mu_{w,G}=\mu_{w',G}$$

for every compact group $G$, then $w$ and $w'$ lie in the same $\operatorname{Aut}(F_r)$-orbit. This is a well-known open converse problem in the theory of word measures and word maps.