Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is the unique group topology such that the set of normal subgroups $N$ with $G/N$ in $W$ is a fundamental system of neighborhoods of the identity.
Let $FG(A)$ be a finitely generated free group. Steinberg and Auinger proved that the pro-$W$ closure of a finitely generated subgroup $H$ of $FG(A)$ is also finitely generated.
Question. Let $H$ be a finitely generated subgroup of a finitely generated free group $FG(A)$. Is there an algorithm to compute a basis for the pro-$W$ closure $H$ given as input a generating set for $H$?
I really stuck in this problem. Of course this is an open problem. Ribes and Zalesskii provided an algorithm for computing a basis for the pro-$p$ closure of a finitely generated subgroup of a free group.