Tosatti's uniqueness conjecture for irrational nef isotropic classes on K3 surfaces

Let $X$ be a $K3$ surface and let $[\alpha]\in\partial\mathcal{C}$ be a nef class satisfying

$$\int_X\alpha^2=0,$$

and

$$\mathbb{R}[\alpha]\cap H^2(X,\mathbb{Q})=\{0\}.$$

Tosatti's uniqueness conjecture. The class $[\alpha]$ contains a unique closed positive $(1,1)$-current.

The source introduces this as a conjecture about irrational nef classes that are not big on $K3$ surfaces; no resolution is stated.