The homological detection conjecture for incompressible surfaces in 3-manifolds

Let $S$ be an incompressible surface in a $3$--manifold $N$, and let $M$ be the manifold obtained by cutting $N$ along $S$. Let $i:S\to M$ be either the positive or the negative inclusion. For a homomorphism $\alpha:\pi_1(N)\to G$ to a finite group, let $H_1^\alpha(-;\mathbb Z[G])$ denote the corresponding twisted homology.

Homological detection conjecture. If

$$i:H_1^\alpha(S;\mathbb Z[G])\to H_1^\alpha(M;\mathbb Z[G])$$

is an isomorphism for every homomorphism $\pi_1(N)\to G$ with $G$ finite, then

$$i:\pi_1(S)\to\pi_1(M)$$

is surjective.

The question is identified as a group-theoretic condition from which the preceding fiber-detection conjecture would follow, together with the Geometrization Conjecture. It remains open in the source.