The abundance conjecture for nef line bundles on Calabi–Yau manifolds

Let $X$ be a simply connected compact Kähler manifold with $c_1(X)=0$. Let $\mathcal L$ be a nef line bundle on $X$.

Abundance conjecture. $\mathcal L$ is semiample, i.e. there exists a positive integer $m$ such that $\mathcal L^{\otimes m}$ is generated by global sections.

This is a standard abundance prediction for simply connected Ricci-flat compact Kähler manifolds. The paper proves a special case for Calabi–Yau threefolds with Picard number $2$, nonzero Euler characteristic, and $c_1(\mathcal L)^2\ne0$; the general statement remains open.