Kollár's local fundamental group conjecture for klt singularities

Let $(X,\Delta)$ be a log pair over $\mathbb C$, and let $x\in(X,\Delta)$ be a Kawamata log terminal (klt) singularity. For a sufficiently small Euclidean ball $B$ around $x$ in $X$, define the local fundamental group

$$\pi_1^{\rm loc}(X,x):=\pi_1(B\setminus\{x\})=\pi_1({\rm Link}(x)),$$

where ${\rm Link}(x)=\partial B$.

Kollár's local fundamental group conjecture. The group $\pi_1^{\rm loc}(X,x)$ is finite.

The conjecture concerns the topology of the smooth neighbourhood of a klt singularity. The source paper proves it, in fact establishing the stronger finiteness of the fundamental group of the smooth locus of a neighbourhood of $x$; it also proves finiteness of the corresponding orbifold fundamental group for weakly Fano pairs.