Conjecture on dimension of Bernoulli convolutions from Pisot conjugates

Let $\nu_{\beta_1}$ denote the Bernoulli convolution associated with $\beta_1$. Suppose that $\beta_1\in(1,2)$, $\beta_1\neq(1+\sqrt{5})/2$, and that $\beta_1$ has a real conjugate $\beta_2$ such that $1/|\beta_2|$ is a Pisot number. Dimension conjecture. We conjecture that

$$\dim(\nu_{\beta_1})=1.$$

This concerns parameters whose reciprocal conjugate is Pisot and predicts full Hausdorff dimension for the corresponding Bernoulli convolution. The supplied text gives no resolution status.