Andrews–Petsche conjecture on abelian arboreal Galois images

Let $K$ be a number field, let $K^{\mathrm{ab}}$ be its maximal abelian extension, let $f:{\mathbb P}^1\to{\mathbb P}^1$ be a polynomial map of degree $d\geq2$ defined over $K$, and let $\alpha\in{\mathbb P}^1(K)$ be non-exceptional for $f$. Andrews–Petsche conjecture. If $G_{f,\alpha}$ is abelian, then either $(f,\alpha)$ is $\mathrm{PGL}_2(K^{\mathrm{ab}})$-conjugate to $(x^d,\zeta)$ for a root of unity $\zeta$, or it is $\mathrm{PGL}_2(K^{\mathrm{ab}})$-conjugate to $(\pm T_d(x),\zeta+\frac{1}{\zeta})$ for a root of unity $\zeta$. This conjecture classifies the expected polynomial cases with abelian arboreal image; the supplied text attributes it to Andrews–Petsche and also records the stated formulation with Ferraguti–Ostafe–Zannier.