The semipositive representative conjecture for nef and big classes on Calabi-Yau manifolds

Let $X$ be a compact Kähler Calabi-Yau manifold, and let $\alpha\in H^{1,1}_{\mathbb{R}}(X)$ be a class that is nef and big but not Kähler. A smooth $(1,1)$-form representing $\alpha$ is pointwise nonnegative and Kähler outside a proper analytic subvariety $E\subset X$. Semipositive representative conjecture. The class $\alpha$ can be represented by such a form $\omega$. This would extend the semipositivity result used in the projective case to arbitrary compact Kähler Calabi-Yau manifolds, providing a key step toward understanding limits of Ricci-flat metrics as Kähler classes degenerate. The source does not indicate that the conjecture has been resolved.