Tosatti's transcendental basepoint-free conjecture for Calabi–Yau manifolds

Let $X$ be a compact Kähler manifold. A class $\alpha \in H^{1,1}(X,\mathbb{R})$ is semi-ample if there exists a holomorphic contraction $\rho:X\to X'$ to a normal analytic space $X'$ and a Kähler class $\omega\in H^{1,1}_{\mathrm{BC}}(X')$ such that $\rho^*\omega=\alpha$. Tosatti's transcendental basepoint-free conjecture. If $X$ is a Calabi–Yau manifold and $\alpha\in H^{1,1}(X,\mathbb{R})$ is big and nef, then $\alpha$ is semi-ample. This conjecture is the transcendental analogue of the basepoint-free theorem for line bundles. The paper proves it conditionally from the corresponding statement for the hyperkähler factors in the Beauville–Bogomolov decomposition and establishes it for certain big and nef classes on hyperkähler manifolds, but the general conjecture remains open.