Free-product conjecture for cancellative rational-map semigroups

Let $S$ be a countable cancellative semigroup of rational maps. A cancellative semigroup is both left and right cancellative, and an amenable subsemigroup admits both left and right invariant means.

Cancellative free-product conjecture. The semigroup $S$ is a free product of its amenable subsemigroups.

This is presented as a version of the preceding finite-index free-product conjecture for cancellative semigroups. The source gives no resolution.