Let $M^3$ be a compact, orientable, irreducible 3-manifold with incompressible boundary. Let $S\subseteq\partial M$ be one of its boundary components. Does $\pi_1(M)$ induce the full profinite topology on the subgroup $\pi_1(S)$? Namely, for any finite index subgroup $H\le \pi_1(S)$, does there exist a finite index subgroup $K\le \pi_1(M)$ such that $K\cap \pi_1(S)\subseteq H$?
This is equivalent to say that for any finite cover $\widetilde{S}$ of $S$, does there exist a finite cover $p: \widetilde{M}^\ast \to M$ such that a component $p^{-1}(S)\to S$ factors through $\widetilde{S}$?
Note that Long-Niblo showed that $\pi_1(S)$ is separable in $\pi_1(M)$, i.e. closed in the profinite topology. So my question is also equivalent to whether every finite index subgroup of $\pi_1(S)$ is also separable in $\pi_1(M)$.
The result holds when the boundary of $M$ is a collection of tori, based on the JSJ-decomposition and the finite covers constructed by Hamilton.
