The Calabi–Yau type conjecture for varieties with polarized endomorphisms

Let $X$ be a normal complex projective variety admitting a non-invertible polarized endomorphism.

Calabi–Yau type conjecture. Then $X$ is of Calabi–Yau type.

This is presented as another important conjecture concerning polarized endomorphisms. The supplied status evidence resolves it under the additional assumptions that $X$ is $\mathbb{Q}$-Gorenstein and the endomorphism is étale in codimension one; the general statement remains open.