Exceptional maps and abelian points in backward orbits

Consider a number field $K$, a polarized endomorphism $f:X\to X$ on a normal projective variety, and a non-exceptional point $α$ for $f$. We show that the extension $K(f^{-\infty}(α))/K$ generated by the backward orbit of $α$ is virtually abelian if and only if $f$ is an exceptional map with CM and