The finite-group word-measure orbit conjecture of Shalev

Let $\mathbf{F}$ be the free group, and let $w_{1},w_{2}\in\mathbf{F}$. For a finite group $G$, write $\mu_{w}$ for the probability measure on $G$ obtained by evaluating $w$ on a Haar-random homomorphism from $\mathbf{F}$ to $G$. Shalev's conjecture. If $\mu_{w_{1}}=\mu_{w_{2}}$ for every finite group $G$, then

$$w_{2}\in\operatorname{Aut}(\mathbf{F}).w_{1}.$$

This is the finite-group strengthening of the compact-group word-measure question. The source gives no evidence of a resolution, so it remains open there.