The boundedness of the index conjecture for klt log Calabi–Yau pairs

Let $d$ be a positive integer and let $\Lambda$ be a set of rational numbers satisfying the descending chain condition. A projective $d$-dimensional klt log Calabi–Yau pair is a projective klt pair $(X,B)$ with $K_X+B\sim_{\mathbb Q}0$, and the coefficients of $B$ lie in $\Lambda$. Boundedness of the index conjecture. There exists a constant $I\coloneqq I(\Lambda,d)$ such that, for every such pair $(X,B)$,

$$I(\Lambda,d)(K_X+B)\sim 0.$$

This is the boundedness-of-index conjecture cited in the paper as a conjecture of Birkar. It is a key input for controlling indices of log Calabi–Yau pairs of higher coregularity, and remains open in the stated generality.