Bounded canonical index conjecture for semi-log canonical Calabi–Yau pairs

Let $d$ be a natural number and let $\mathfrak{R}\subset [0,1]$ be a finite set of rational numbers. For a finite set $\mathfrak{R}$, write

$$\Phi(\mathfrak{R})=\left\{1-\frac{r}{m}\mid r\in\mathfrak{R},\ m\in\mathbb{N}\right\}\cup\{1\}.$$

A pair $(X,B)$ is semi-log canonical if it has the standard semi-log canonical singularities. Bounded canonical index conjecture. There exists a natural number $n$, depending only on $d$ and $\mathfrak{R}$, such that whenever $(X,B)$ is a projective semi-log canonical pair of dimension $d$, $B\in\Phi(\mathfrak{R})$, and $K_X+B\sim_{\mathbb Q}0$, one has

$$n(K_X+B)\sim 0.$$

This predicts a uniform bound for the canonical index of semi-log canonical log Calabi–Yau pairs; the paper establishes related boundedness results in dimension two, but the stated general conjecture remains open.