Uniform potential convergence conjecture for boundary classes on K3 surfaces

Let $X$ be a $K3$ surface, let $\alpha$ be a closed real $(1,1)$-form with $0\neq[\alpha]\in\partial\mathcal{C}_X$ and $\int_X\alpha^2=0$, and let $\omega$ be a Kähler metric on $X$. For $0<t\leqslant 1$, write

$$\omega_t=\alpha+t\omega+\sqrt{-1}\,\partial\overline{\partial}\varphi_t$$

for the Ricci-flat Kähler metric in the class $[\alpha]+t[\omega]$, normalized by $\sup_X\varphi_t=0$. Uniform potential convergence conjecture. There is a closed positive $(1,1)$-current

$$\beta=\alpha+\sqrt{-1}\,\partial\overline{\partial}\varphi_0\geqslant 0$$

with $\varphi_0\in C^0(X)$ and $\sup_X\varphi_0=0$, such that

$$\varphi_t\to\varphi_0$$

uniformly on $X$ as $t\to 0$.

The conjecture is known for eigenclasses, where the limiting potential is Hölder continuous. It remains open for classes from elliptic fibrations and, most notably, for classes that are neither eigenclasses nor pulled back from an elliptic-fibration base.