Let $G = \langle a,b,c\:|\: a^2, b^2, c^2\rangle$. Let $\tilde{}$ by the equivalence relation on $G$ generated by conjugation and inversion (i.e., $x\tilde{} y$ if there is a finite sequence of conjugations and inversions which transform $x$ into $y$). Let $x,y\in G$ and suppose there exists a $z\in G$ such that $xzy^{-1}z^{-1}$ is a commutator. Is $x\tilde{} y$?
This question arises in the study of triangular billiards. Specifically, I am interested in a subgroup of $G$ and elements $z$ of a certain form, and there is a great deal of computational evidence that the statement holds in this case. However, the subgroup admits no simple description outside the theory of billiards, and I am hoping that I can simply hammer the statement with a combinatorial proof for the whole group $G$.