Let $F = \langle a,b \rangle$ be a non-abelian free group.

Question:Is there an algorithm that takes as input $x,y,z \in F$ and answers the question whether $x$ is a product of conjugates of $y$ and $z$, i.e. whether there exists $g,h \in F$ with $$ x = gyg^{-1} \cdot hzh^{-1} ?$$

It is obvious that there exists an algorithm that enumerates all products of conjugates of $y$ and $z$. What is missing is a certificate that $x$ cannot be written in this form. Sometimes (like in the case of the word problem or the membership problem for finitely generated subgroups), this part is done by looking at the finite quotients and finding one finite quotient so that $x$ is not in the product of the conjugacy classes. Hence, one starting point would be to ask if the product of conjugacy classes is closed in the pro-finite topology. This was conjectured by Stallings to hold even in the pro-p topology and disproved (in the pro-p case) by Howie in

The p-adic Topology on a Free Group: A Counterexample, Math. Z. 187,25-27 (1984)

It might hold in the pro-finite topology, but this seems to be difficult. Anyhow, I am just looking for an algorithm, maybe this can be done without looking at the finite quotients.