Incubulable hyperbolic 3-pseudomanifold groups

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental groups have property (T). In particular, the fundamental groups of these 3-pseudomanifolds are word hyperbolic but not cubulable. We deduce that in any relative cubulation of one of these hyperbolic 3-manifold groups some hyperplane stabilizer has infinite intersection with the fundamental group of some boundary component.