Milnor's Hausdorff-dimension conjecture for exceptional hyperbolic component boundaries

Let $\mathcal F$ be an algebraic family in the space $\operatorname{Rat}_d$ of rational maps of degree $d\geq 2$, and let $\mathcal H$ be a hyperbolic component in $\mathcal F$. Call $\mathcal H$ exceptional if all maps on $\partial\mathcal H$ have at least one indifferent cycle. Write $\operatorname{H.dim}(\partial\mathcal H)$ for the Hausdorff dimension of $\partial\mathcal H$, and let $\dim_{\mathrm{top}}(\partial\mathcal H)$ denote its topological dimension.

Milnor's Hausdorff-dimension conjecture. If $\mathcal H$ is not exceptional, then $\partial\mathcal H$ is a fractal set in the sense that

$$\operatorname{H.dim}(\partial\mathcal H)>\dim_{\mathrm{top}}(\partial\mathcal H),$$

where $\dim_{\mathrm{top}}(\partial\mathcal H)$ is the topological dimension of $\partial\mathcal H$.

The conjecture concerns the analytic complexity of non-exceptional hyperbolic component boundaries, asserting that their Hausdorff dimension strictly exceeds their topological dimension. The source presents this as Milnor's Conjecture 2(b); no resolution is given.