Sharing a measure of maximal entropy in polynomial semigroups

Let $P_1,P_2,\dots, P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or to Chebyshev polynomials, and $S$ the semigroup under composition generated by $P_1,P_2,\dots, P_k$. We show that all elements of $S$ share a measure of maximal entropy if and only if the intersection of principal right ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.