Tosatti's bounded-potential conjecture for nef classes

Let $X$ be a Calabi–Yau manifold and let $[\alpha]\in\overline{\mathcal{C}}$ be a nef class. A closed positive current in $[\alpha]$ is a current of the form $T=\alpha+i\partial\overline{\partial}\varphi\geqslant0$ with $\varphi$ bounded.

Tosatti's bounded-potential conjecture. There is $\varphi\in L^\infty(X)$ such that

$$T=\alpha+i\partial\overline{\partial}\varphi\geqslant0$$

is a closed positive current in $[\alpha]$.

The conjecture is open even on $K3$ surfaces for nef classes that are not big, although the source records partial results for certain projective $K3$ surfaces.