Bifurcation-set equality conjecture for beta transformations with a hole

Let $\beta\in(1,2]$. The bifurcation-set equality conjecture.

$$\mathcal{B}_\beta=\mathcal{E}_\beta$$

if and only if $\mathcal{E}_\beta$ has no isolated points. Here $\mathcal{E}_\beta$ is the set-valued bifurcation set and $\mathcal{B}_\beta$ is the Hausdorff-dimension bifurcation set associated with the beta transformation with a hole at $0$. The conjecture proposes that equality of these two bifurcation sets is equivalent to the absence of isolated points in the set-valued bifurcation set; its resolution is not supplied in the source.