Pilgrim's finite curve attractor conjecture

Let $f\colon \widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ be a post-critically finite rational map that is not a flexible Lattès map. A simple closed curve on $(\widehat{\mathbb{C}},P_f)$ is called essential when it is essential in the punctured sphere, and $f^{-n}(\gamma)$ denotes the full preimage of a curve under the $n$-fold iterate of $f$. Pilgrim's finite curve attractor conjecture. There exists a finite set $X$ of homotopy classes of essential simple closed curves on $(\widehat{\mathbb{C}},P_f)$ such that, for any simple closed curve $\gamma$ on $(\widehat{\mathbb{C}},P_f)$, every essential component of $f^{-n}(\gamma)$ belongs to $X$ for all sufficiently large $n>0$. This conjecture describes a finite attractor for the pullback dynamics of curves associated with a post-critically finite rational map; the supplied source does not state whether it has been resolved.