The global curve attractor problem for rational Thurston maps

Let $f\colon S^2\to S^2$ be a Thurston map with a hyperbolic orbifold that is realized by a rational map. Let $P_f$ denote its postcritical set, and let pullbacks be considered up to isotopy relative to $P_f$. Global curve attractor problem. There exists a finite set $\mathscr{A}(f)$ of Jordan curves in $S^2\setminus P_f$ such that, for every Jordan curve $\gamma\subset S^2\setminus P_f$, all pullbacks $\widetilde{\gamma}$ of $\gamma$ under $f^n$ belong to $\mathscr{A}(f)$ up to isotopy relative to $P_f$ for all sufficiently large $n\in\mathbb{N}$. This is a difficult open question in holomorphic dynamics, concerning whether pullbacks of every Jordan curve are eventually captured by a finite global attractor of isotopy classes.