The non-manifold 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.

Non-manifold conjecture.

$$AH(M)\text{ is not a manifold.}$$

The paper presents this as an easier conjecture, especially when $M$ does not contain a primitive essential annulus. The source gives no resolution, so the conjecture remains open.