Dimension-one conjecture for non-golden-ratio Pisot reciprocal conjugates

Let $u_{\beta_1}$ be the Bernoulli convolution associated with $\beta_1\binom{}{}$ and let $\beta_2$ be a real conjugate of $\beta_1$. Assume that $\beta_1\text{ lies in }(1,2)$, $\beta_1 eq(1+\frac{1}{})$ and $1/|\beta_2|$ is a Pisot number.

Dimension-one conjecture. Under these assumptions,

$$\operatorname{dim_H}(\nu_{\beta_1})=1.$$

The surrounding text presents this as an observed consequence for the listed algebraic integers, while the candidate itself is not accompanied by a resolution status.