Bromberg's local-connectivity conjecture for surface-group deformation spaces

Let $S$ be any compact surface. Write $AH(S\times I,\partial S\times I)$ for the deformation space of marked hyperbolic structures on $S\times I$ with the indicated parabolic locus. A space is locally connected if every point has arbitrarily small connected neighborhoods.

Bromberg's surface-group conjecture.

$$AH(S\times I,\partial S\times I)\text{ is not locally connected.}$$

The paper notes that this was proved by Aaron Magid when $S$ is a closed orientable surface of genus at least two. The unrestricted statement for every compact surface is therefore not left open by the source's stated evidence in those cases, but the candidate does not provide a complete resolution for all compact surfaces.