The smooth convergence conjecture for collapsing Ricci-flat metrics

In Setup I.B.1, let $f:X\to Y$ be the fiber space and let $S$ be the singular-fiber locus. Smooth convergence conjecture. The Ricci-flat Kähler metrics satisfy

$$\omega_t\to f^*\omega_0$$

in $C^\infty_{\operatorname{loc}}(X\setminus S,\omega_X)$. The known result gives convergence of Kähler potentials only in $C^{1,\gamma}_{\operatorname{loc}}$ for every $\gamma<1$, together with local metric comparability; improving this to smooth local convergence is the stated open problem.