Invariant maximal-entropy measure conjecture for rational-map semigroups

Let $S$ be a semigroup of rational maps. An invariant probability measure is a probability measure invariant under every element of $S$, and a non-injective element is a map in $S$ that is not injective. Invariant maximal-entropy measure conjecture. If $S$ admits an invariant probability measure that is the measure of maximal entropy of every non-injective element of $S$, then $S$ is right-amenable. This would extend the corresponding characterization known for semigroups generated by finite non-exceptional collections of polynomials to rational-map semigroups; the general rational-map case is stated as an open question in the source.