For a finitely presented group $G$, generated by a finite set $A$, the *commutator problem* is the decision problem: given a word $w$ over the alphabet $A \cup A^{-1}$, can one decide if $w$ is a commutator, i.e. whether there exist words $x, y$ such that $w = [x, y]$ in $G$. Here $[x, y] = x^{-1}y^{-1}xy$ is the commutator. The commutator problem was solved for free groups by Wicks in 1962.

In 1981, Comerford & Edmunds [1] asked whether decidability of the commutator problem for $G$ implies decidability of the conjugacy problem, or even the word problem, for $G$. **Has there been any recent progress on this question since then, or any results in a similar direction?**

[1] *Comerford, Leo P. jun.; Edmunds, Charles C.*, **Quadratic equations over free groups and free products**, J. Algebra 68, 276-297 (1981). ZBL0526.20024.