Milnor's non-local-connectivity conjecture for 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 $\mathcal F$. A hyperbolic component is of UGO type if the basin of every attracting cycle contains exactly one grand orbit equivalence class of critical points. A map has a free critical point when the critical point is not identified with another critical point under the relevant grand-orbit equivalence.

Milnor's non-local-connectivity conjecture. If the maps in $\mathcal H$ have an attracting cycle with two distinct free critical points in its attracting basin, then $\partial \mathcal H$ is not locally connected.

Milnor proved that UGO hyperbolic components have semi-algebraic, hence locally connected, boundaries. The conjecture predicts the opposite behavior when an attracting basin contains two distinct free critical points.