Transcendental base-point-free conjecture for nef classes

Let $X$ be a compact Kähler manifold and let $[\alpha]$ be a nef $(1,1)$ class on $X$. Suppose that $\lambda[\alpha]-c_1(K_X)$ is nef and big for some $\lambda>0$. Transcendental base-point-free conjecture. The class $[\alpha]$ is semiample. This is the transcendental analogue of the Kawamata–Shokurov base-point-free theorem; related formulations are known for Calabi–Yau manifolds and weaker versions, but the stated general conjecture is unresolved.