Characterization of Chebyshev-like maps by affine Weyl iterated monodromy groups

Let $f: \mathbb{C}^n \to \mathbb{C}^n$ be post-critically finite. An affine Weyl iterated-monodromy characterization conjecture asserts that if the iterated monodromy group of $f$ is an affine Weyl group, then some iterate of $f$ is a Chebyshev-like map.

The conjecture proposes a converse to the theorem that Chebyshev-like maps associated to root systems have iterated monodromy groups isomorphic to the corresponding affine Weyl groups. It would characterize these maps among post-critically finite maps through their iterated monodromy groups.