Broustet–Gongyo's Calabi–Yau type conjecture for polarized endomorphisms

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, meaning that $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. A variety $X$ is of Calabi–Yau type if there is an effective $\mathbb{Q}$-divisor $\Delta$ such that $(X,\Delta)$ is log canonical and

$$K_X+\Delta\sim_{\mathbb{Q}}0.$$

Broustet–Gongyo's conjecture. Every normal projective variety admitting a polarized endomorphism is of Calabi–Yau type. This has been proved for surfaces, smooth threefolds, and rationally connected smooth projective varieties, but remains open in general.