The log-canonical model conjecture for varieties admitting endomorphisms

Let $X$ be a normal variety, and let $f\colon X\rightarrow X$ be an endomorphism of degree $\deg(f)>1$. Suppose that $X$ admits a log-canonical model

$$\mu\colon Y\rightarrow X.$$

Let $E^{lc}_\mu$ be the sum of all the $\mu$-exceptional prime divisors, each with coefficient one, and let $Z$ be an irreducible component of $\mu(E^{lc}_\mu)$. The log-canonical model conjecture. Up to replacing $f$ by some iterate, $Z$ is totally invariant. In this case, $Z$ is not contained in the ramification divisor $R$, and the induced endomorphism $f|_Z\colon Z\rightarrow Z$ satisfies

$$\deg(f|_Z)=\deg(f).$$

If moreover $X$ is projective and $f$ is polarised, then $\mu$ is an isomorphism in codimension one. The statement concerns how endomorphisms interact with log-canonical models and their exceptional divisors. The surrounding discussion indicates that the existence of log-canonical models is known for certain pairs, while existence in general would follow from the minimal model program, including abundance; the status of this specific assertion is not resolved in the supplied text.