Tosatti's minimal-singularities conjecture for degenerate Ricci-flat limits

Let $X$ be a Calabi–Yau manifold, let $[\alpha]\in\overline{\mathcal{C}}$ be a nef class, and let $\{[\alpha_i]\}_{i\geqslant0}\subset\mathcal{C}$ satisfy $[\alpha_i]\to[\alpha]$. Let $\mathcal{T}_{[\alpha]}$ denote the relevant space of limiting currents in the class $[\alpha]$. A current has minimal singularities in $[\alpha]$ if its potential is less singular than the potential of every other closed positive current in that class.

Tosatti's minimal-singularities conjecture. If $\eta\in\mathcal{T}_{[\alpha]}$ and

$$\mathrm{CY}([\alpha_i])\to\eta$$

in the weak topology, then $\eta$ has minimal singularities in the class $[\alpha]$.

The conjecture is known for sequences $[\alpha_i]=[\alpha]+\varepsilon_i[\omega]$ with $\varepsilon_i>0$ and $\varepsilon_i\to0$, but remains open for more general sequences.