Finite-index free-product conjecture for rational-map semigroups

Let $S$ be a countable semigroup of rational maps. A right amenable subsemigroup is a subsemigroup admitting a right invariant mean, and a subsemigroup has finite index in $S$ in the sense intended by the source.

Finite-index free-product conjecture. The semigroup $S$ contains a finite-index subsemigroup which is a free product of its right amenable subsemigroups.

The conjecture is motivated by embedding results for cancellative rational-map semigroups and the structure of semigroups without free rank-two subsemigroups. The source gives no resolution.