Periodic eigenvalue conjecture for post-critically finite endomorphisms of projective space

Let $f \colon {\mathbb{CP}}^n \to {\mathbb{CP}}^n$, with $n \ge 2$, be a post-critically finite endomorphism of degree $d \ge 2$, and let $\lambda$ be an eigenvalue of $f$ along a periodic cycle. Periodic eigenvalue conjecture. Then either

$$\lambda=0$$

or

$$|\lambda|>1.$$

This conjecture extends the classical one-dimensional result to higher-dimensional projective space. It remains open in general, although the paper verifies it for weakly post-critically finite all the way down maps, including Koch maps.