A 3-manifold with a non-subgroup-separable fundamental group

We examine a 3-manifold Gamma whose fundamental group is known to be non-subgroup-separable (non-LERF). We show that this manifold Gamma is a graph manifold and that the subgroup known to be non-separable is not geometric. On the other hand, there are incompressible surfaces immersed in the manifold which do not lift to embeddings in any finite-degree covering space. We then prove that these bad incompressible surfaces must have non-empty boundary.