Finiteness conjecture for the fundamental group of the regular locus of ICY and IHS varieties

Let $X$ be an ICY or IHS variety, and let $X_{\rm reg}$ denote its regular locus.

Fundamental-group finiteness conjecture. The fundamental group

$$\pi_1(X_{\rm reg})$$

is finite.

This is identified as the major remaining open question from a topological point of view in the development of the Beauville–Bogomolov decomposition for klt varieties. The conjecture asks whether the regular loci of both irreducible Calabi–Yau and irreducible holomorphic symplectic varieties have finite fundamental group.