Conjecture on stable basins in forward orbits of transcendental semigroups

Let $S$ be a finitely generated transcendental semigroup, let $F(S)$ be its Fatou set, and let $U$ be a component of $F(S)$. A stable basin $V$ for $S$ is a basin lying in the forward orbit of $U$ whose stabilizer has cofinite index in $S$.

Forward-orbit stable-basin conjecture. For each component $U$ of $F(S)$, there is a stable basin $V$ for $S$ lying in the forward orbit of $U$ that has cofinite index in $S$.

This is presented as a second analogue, for transcendental semigroups, of conjectures for rational semigroups; the source notes that the analogous statements are true in classical complex dynamics. No resolution is supplied here.