Conjecture on cofinite-index stable basins in transcendental semigroup dynamics

Let $S$ be a finitely generated transcendental semigroup such that $F(S)\neq\emptyset$. A stable basin $U$ is a basin whose stabilizer has cofinite index in $S$.

Stable-basin conjecture. Then a stable basin $U$ has cofinite index in $S$.

This is proposed as an analogue for transcendental semigroups of conjectures for rational semigroups, and the statement is noted to hold in classical complex dynamics. Its resolution for finitely generated transcendental semigroups is not given.