Remeslenikov's conjecture on profinite rigidity of finitely generated free groups

Let $F$ be a finitely generated free group. Two groups have the same finite quotients when their profinite completions are isomorphic.

Remeslenikov's conjecture. Finitely generated free groups are profinitely rigid: every finitely generated, residually finite group with the same finite quotients as $F$ is isomorphic to $F$.

This is presented as the most celebrated problem concerning profinite rigidity, and the answer is widely believed to be positive. The source points to discussion of related results and recent progress, but does not state that the conjecture has been resolved.