The Douady–Sullivan criterion conjecture for non-locally-connected Fatou boundaries

Let $f$ be a rational map whose Julia set is connected. A periodic Fatou component is a Fatou component eventually returning to itself under iteration, and a boundary is locally connected if every point has arbitrarily small connected neighborhoods in the boundary. Douady–Sullivan criterion conjecture. If $f$ has a periodic Fatou component which contains a critical point whose boundary is not locally-connected, then $f$ satisfies the Douady–Sullivan criterion. This proposes a dynamical obstruction to local connectivity for rational maps, including polynomials, motivated by examples where critical periodic Fatou components have non-locally-connected boundaries; the source presents it as an unproved conjecture.