The conjecture that word measures separate automorphism orbits

Let $\mathbb{\mathbf{F}}$ be a free group and let $w_{1},w_{2}\in\mathbb{\mathbf{F}}$. For each word, consider the induced probability measure on every compact group. Word-measure orbit conjecture. If $w_{1}$ and $w_{2}$ induce the same measure on every compact group, then they belong to the same $\operatorname{Aut}(\mathbb{\mathbf{F}})$-orbit. The forward implication from belonging to the same automorphism orbit to having the same word measure is elementary, but the converse is presented as an open conjecture suggested by several mathematicians.