Generalized log-Calabi–Yau conjecture for polarized endomorphisms

Let $X$ be a normal projective variety, let $f\colon X\to X$ be a $q$-polarized endomorphism, and let $\Delta$ be an effective $\mathbb{Q}$-divisor such that $(X,\Delta)$ is an $f$-pair, meaning that

$$R_{\Delta}:=\Delta+R_f-f^*\Delta\geq 0,$$

where $R_f$ is determined by $K_X=f^*K_X+R_f$. Generalized log-Calabi–Yau conjecture. After replacing $f$ by an iterate, both

$$K_X+\Delta+\frac{R_{\Delta}}{q-1}$$

is $\mathbb{Q}$-Cartier and numerically trivial, and

$$\left(X,\Delta+\frac{R_{\Delta}}{q-1}\right)$$

is log canonical. This extends the preceding conjectures to $f$-pairs; the source presents it as a conjecture, and its general status is open.