Large-degree Pisot Bernoulli convolution dimension conjecture

Given a number $\beta\in(1,2)$, let $\nu_{\beta}$ denote the associated Bernoulli convolution and let $\dim_H$ denote Hausdorff dimension. Large-degree Pisot dimension conjecture. If $(\beta_n)$ is a sequence of Pisot numbers in $(1,2)$ whose degrees tend to infinity, then

$$\dim_H(\nu_{\beta_n})\to 1.$$

Numerical evidence suggests that Bernoulli convolutions associated with Pisot numbers of large degree have dimension close to one. The conjecture concerns the limiting dimension in this regime and is presented as an open conjecture in the source.