Hausdorff dimension conjecture for attractors of non-quadratic Pisot numbers

Let $\beta$ be a Pisot number, meaning an algebraic integer greater than $1$ whose other Galois conjugates have modulus less than $1$, and let $A_\beta$ denote the associated attractor. If $\beta$ is not quadratic, then Hausdorff-dimension conjecture.

$$0<\dim_H(A_\beta)<1.$$

This predicts that the attractor has strictly positive but non-full Hausdorff dimension for every non-quadratic Pisot number. The source presents this as a conjecture motivated by the finiteness of the associated transition graph, and no resolution is given.