Tosatti's diameter-collapse conjecture for irrational nef isotropic K3 classes

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

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

and

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

For a class $[\omega]\in\mathcal{C}$, let $\mathrm{CY}([\omega])$ denote the associated Ricci-flat metric.

Tosatti's diameter-collapse conjecture. For every sequence $[\alpha_i]\in\mathcal{C}$ with $[\alpha_i]\to[\alpha]$,

$$\mathrm{diam}(X,\mathrm{CY}([\alpha_i]))\to0.$$

If true, the Gromov–Hausdorff limit of the corresponding metric spaces would be a point. The source gives no resolution of the conjecture.