The incompressible-torus obstruction to L-spaces

Let $Y$ be an irreducible homology sphere, and suppose that it contains an incompressible torus, meaning an embedded torus whose inclusion induces an injective map on fundamental groups. Incompressible-torus obstruction conjecture. If $Y$ contains an incompressible torus, then $Y$ is not an L-space. This conjecture would reduce the L-space homology-sphere classification conjecture to the case of hyperbolic 3-manifolds. Its general status is open.