The profinite rigidity conjecture for automorphic orbits of free-group words

Let $F_k$ be the free group of rank $k$, let $\widehat{F}_k$ be its profinite completion, and let $w,w'\in F_k$. An automorphism of $\widehat{F}_k$ acts on the embedded elements $w,w'$.

Profinite rigidity conjecture for word automorphic orbits. If there exists $\widehat{f}\in\operatorname{Aut}(\widehat{F}_k)$ such that

$$\widehat{f}(w)=w',$$

then there exists $f\in\operatorname{Aut}(F_k)$ with

$$f(w)=w'.$$

This is an equivalent formulation of the long-standing conjecture that two words inducing the same measure on every finite group must lie in the same $\operatorname{Aut}(F_k)$-orbit. The source presents the claim as a conjecture and gives no resolution.