Finite global attractor conjecture for pullbacks of curves under rational Thurston maps

Let $f$ be a rational Thurston map, and consider the pullback relation on nontrivial curves associated to $f$. A finite global attractor conjecture asserts that, if $f$ is not a flexible Lattès example, then the pullback relation on curves has a finite global attractor.

This conjecture predicts finite asymptotic complexity for iterated pullbacks of curves for rational Thurston maps outside the flexible Lattès family. The supplied text does not state a resolution, so its status remains open.