Finiteness conjecture for fundamental groups of klt Calabi–Yau varieties

Let $Y$ be a projective klt variety with numerically trivial canonical divisor, meaning that $K_Y$ is numerically equivalent to zero. Its augmented irregularity is the supremum of the irregularities of finite quasi-étale covers of $Y$; vanishing augmented irregularity means that this invariant is zero. Finiteness conjecture. If $Y$ has vanishing augmented irregularity, then

$$\pi_1(Y)\text{ is finite}.$$

This conjecture reduces a uniformization problem for klt pairs with nef anti-log canonical divisor to the fundamental groups of Calabi–Yau varieties. It is presented as a remaining conjectural step in the paper.