Shokurov's index conjecture for singularities

Let $n\in\mathbb{Z}_{>0}$ and $a\in\mathbb{R}_{\geq 0}$. For an $n$-dimensional $\mathbb{Q}$-Gorenstein variety $X$ and a closed point $p\in X$, write $\operatorname{mld}_p(X)$ for the minimal log discrepancy at $p$, and let the Cartier index of $K_X$ at $p$ be the least positive integer $r$ such that $rK_X$ is Cartier near $p$.

Shokurov's index conjecture. There exists a positive integer $r(n,a)$ such that, whenever $\operatorname{mld}_p(X)=a$, the Cartier index of $K_X$ at $p$ is at most $r(n,a)$.

The paper proves this conjecture for quotient singularities; the general statement is attributed to Shokurov.