The finite-components conjecture for deformation spaces

Let $M$ be a compact hyperbolizable $3$-manifold, and let $AH(M)$ be its deformation space of marked hyperbolic structures. Its interior is denoted by ${\rm \int}(AH(M))$, and a space has finitely many components when its connected-component set is finite.

Finite-components conjecture.

$$AH(M)\text{ has finitely many components if and only if its interior has finitely many components.}$$

The paper proposes this as a simpler step toward enumerating the components of $AH(M)$ in the compressible-boundary case. It is stated without a resolution and remains open.