Suppose $G$ is a finitely generated group. $G_n = \langle\{g^n| g \in G\}\rangle$. Suppose $p$ and $q$ are coprime integers. Is the index of $G_{pq}$ in $G_p \cap G_q$ always finite?
I have recently asked a similar question about arbitrary groups and received an infinitely generated counterexample: Is $[G_p \cap G_q:G_{pq}]$ always finite?
However, the question, whether the statement is true under this additional condition, or is there a finitely generated counterexample, seems to be quite interesting and still remains unanswered.