Kollár's almost abelianity conjecture for projective manifolds with Kodaira dimension zero

Let $X$ be an $n$-dimensional projective manifold. The notation $\kappa(X)$ denotes its Kodaira dimension, and a group is almost Abelian if it contains an Abelian subgroup of finite index.

Kollár's conjecture. If

$$\kappa(X)=0,$$

then

$$\pi_1(X)$$

is almost Abelian.

Kollár's conjecture is known in complex dimensions three and four, but is largely open in higher dimensions. In the source it is presented as a consequence of the preceding conjecture assuming the existence of good minimal models.