Invariant maximal-entropy measure conjecture for free-subsemigroup-free rational-map semigroups

Let $S$ be a finitely generated semigroup of rational maps, and say that a rank $2$ free subsemigroup is a free subsemigroup generated by two elements. 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 $S$ does not contain a rank $2$ free subsemigroup, then $S$ admits an invariant probability measure that is the measure of maximal entropy of every non-injective element of $S$. The conjecture is motivated by the preceding observation that semigroups containing free two-generated subsemigroups are neither right- nor left-amenable; it proposes the existence of a common maximal-entropy measure under the stated obstruction to freeness.