The cellular Markov partition conjecture for expanding Thurston maps

Let $f:S^2\rightarrow S^2$ be an expanding Thurston map. A cellular Markov partition for $f$ is a cellular decomposition whose subdivision is compatible with $f$ in the Markov sense.

Cellular Markov partition conjecture. Every expanding Thurston map admits a cellular Markov partition.

The paper notes that sufficiently large iterates of every expanding Thurston map admit two-tile subdivision rules, and conjectures that the map itself should admit a more general cellular Markov partition. The status is not resolved in the supplied text.