The index conjecture for log 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 $\Phi(\mathfrak{R})$ for the associated coefficient set. Let $(X,B)$ be a projective pair of dimension $d$, with coefficients of $B$ in $\Phi(\mathfrak{R})$ and $K_X+B\sim_\mathbb Q0$. The index conjecture for log Calabi–Yau pairs. There exists a natural number $n$, depending only on $d$ and $\mathfrak{R}$, such that if $(X,B)$ is log canonical, then

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

This is the main index-boundedness conjecture for log canonical Calabi–Yau pairs. It is enough to consider coefficients in the finite set $\mathfrak{R}$; the conjecture is a central boundedness question for the Cartier index of numerically or rationally trivial log canonical divisors.