Uniform expansion conjecture for the Julia set of the post-critically finite endomorphism

Let $f$ be the post-critically finite endomorphism of $\mathbb{P}\mathbb{C}^2$ considered above, with Julia set $J$. An orbispace metric is a metric on the orbifold-like space obtained by recording the post-critical structure. Uniform expansion conjecture. There exists an orbispace metric on $\mathbb{P}\mathbb{C}^2\setminus\{[1:0:0]\}$ such that $f$ is uniformly expanding with respect to this metric on a neighborhood of the Julia set.

The conjecture is the expansion property needed to apply techniques from iterated monodromy groups to the combinatorics and topology of $J$; the paper states that its results hold only modulo this conjecture. Its resolution is not specified in the supplied text.