The non-local-connectivity conjecture for hyperbolizable 3-manifold deformation spaces

Let $M$ be a compact hyperbolizable $3$-manifold with a non-toroidal boundary component. Let $AH(M)$ denote its deformation space of marked hyperbolic structures, and call a space locally connected if every point has arbitrarily small connected neighborhoods.

Non-local-connectivity conjecture.

$$AH(M)\text{ is not locally connected.}$$

The paper describes this as a more dramatic conjecture suggested by the surface-group and Bers-slice cases. It is stated as an open problem about the global topology of deformation spaces.