Finite global attractor conjecture for Thurston pullback relations

Let $f\colon (\widehat{\mathbb{C}},P)\to(\widehat{\mathbb{C}},P)$ be a rational Thurston map with a hyperbolic orbifold. Let $\overline{\mathscr{C}_P}$ be the set of isotopy classes of essential and non-essential curves in $\widehat{\mathbb{C}}\setminus P$, with $\odot$ denoting the non-essential class, and let $\xleftarrow{f}$ be the pullback relation on curves. Finite global attractor conjecture. There is a finite set $\mathcal{A}\subset\overline{\mathscr{C}_P}$ such that every orbit

$$[\gamma_0]\xleftarrow{f}[\gamma_1]\xleftarrow{f}[\gamma_2]\xleftarrow{f}\dots$$

eventually lies in $\mathcal{A}$. This conjecture predicts finite global dynamical behavior for pullback on curves of rational Thurston maps with hyperbolic orbifold; the source presents it as a motivation for the paper, and no resolution is supplied in the given text.