Universality conjecture for Hausdorff dimension of cubic critical circle maps

Let $\mu$ be the unique normalized invariant measure of a cubic critical homeomorphism of the circle, and let $HD(\mu)$ denote the Hausdorff dimension of its measure-theoretical support. Consider cubic critical homeomorphisms with rotation number of algebraical degree $2$, and their topological conjugacy classes. Universality conjecture. $HD(\mu)$ is constant in any topological conjugacy class of cubic critical homeomorphisms with rotation number of algebraical degree $2$. The statement is presented as a consequence of the golden mean universality conjecture and is supported by computer-based arguments, but rigorous evidence is scarce; it remains an interesting open question.