Bromberg's conjecture on local connectivity of Bers-slice closures

Let $S$ be a compact hyperbolic surface whose interior $S^0$ is not a once-punctured torus, thrice-punctured sphere, or 4-times punctured sphere. Let $QF(S)$ be the quasifuchsian deformation space, and let a Bers slice be one of its standard slices obtained by fixing one conformal structure at infinity. A closure is locally connected if each point has arbitrarily small connected neighborhoods in the closure.

Bromberg's Bers-slice conjecture. The closure of any Bers slice in $QF(S)$ is not locally connected.

The conjecture is presented as an expected extension of known non-local-connectivity phenomena, in settings where the wrapping construction may not apply. The source gives no resolution, so it remains open.