Higher-genus deformation-space non-local-connectivity conjecture

Let $S$ be a compact surface with boundary. Write $AH(S\times[0,1],\partial S\times[0,1])$ for the deformation space of marked hyperbolic structures on the pared $3$-manifold $(S\times[0,1],\partial S\times[0,1])$. Higher-genus deformation-space conjecture. The space

$$AH(S\times[0,1],\partial S\times[0,1])$$

is not locally connected. This proposes that the non-local-connectivity phenomenon exhibited for punctured-torus groups persists for compact surfaces with boundary; the source gives no proof or resolution.