The compact-group word-measure orbit conjecture

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

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

This asks whether word measures on all compact groups separate the $\operatorname{Aut}(\mathbf{F})$-orbits of words. It is presented as a conjecture in the source, with no resolution stated.