Invariant means on non-exceptional left amenable rational-map semigroups

Let $S$ be a non-exceptional left amenable semigroup of rational maps. A mean on $S$ is left invariant if it is invariant under left multiplication, and right invariant if it is invariant under right multiplication.

Invariant-means conjecture. The set of left invariant means on $S$ coincides with the set of right invariant means on $S$.

This would extend the corresponding equality known for amenable semigroups of polynomials to non-exceptional rational-map semigroups. The source gives no resolution of the conjecture.