Soldatenkov–Verbitsky conjecture for smooth families of holomorphically symplectic manifolds

Let $X_t$ be a smooth family of compact holomorphically symplectic manifolds over a disk $\Delta$. Assume that $X_t$ is hyperkähler for all $t\neq 0$. Soldatenkov–Verbitsky conjecture. The central fiber $X_0$ belongs to Fujiki class $\mathcal C$. This is a precise family version of the conjecture on degeneration limits of hyperkähler manifolds. The paper states a stronger theorem asserting that $X_0$ is Kähler, so this formulation is solved.