Let $F$ be a nonabelian finitely generated free profinite group, and let $x \in F$. Must there be some $1 \neq y \in F$ such that $\langle x,y \rangle$ is isomorphic to the free profinite product of $\langle x\rangle$ and $\langle y \rangle$ ?
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$\begingroup$ It seems plausible that this can be proved using the theory of profinite trees. You should check the papers of Zalesskii and his co-authors. $\endgroup$– HJRWCommented Jan 28, 2015 at 22:12
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