The polynomial-Julia-set accumulation conjecture for critical Fatou boundaries

Let $f$ be a rational map whose Julia set is connected. A periodic critical Fatou component is a periodic Fatou component containing a critical point, and a non-locally-connected polynomial Julia set is the Julia set of a polynomial that is not locally connected. Polynomial-Julia-set accumulation conjecture. If $f$ has a periodic critical Fatou component $U$ whose boundary is not locally-connected, then $\partial U$ contains the homeomorphic image of some non-locally-connected polynomial Julia set. This conjecture is motivated by examples in which boundaries of critical periodic Fatou components contain or are homeomorphic to non-locally-connected quadratic Julia sets; whether this phenomenon holds for all such rational maps remains open in the source.