Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

Let $X$ be a compact hyperk\"ahler manifold with a Lagrangian fibration $\pi\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $\pi^\varphi\colon X^\varphi\to B$ which is isomorphic to $\pi$ locally over the base. In particular, $\pi^\varphi$ has the same fibers as $\pi$. A twist $X^\varphi$ corresponds to an element $\varphi$ in the Shafarevich-Tate group of $X$. We show that $X^\varphi$ is K\"ahler when a multiple of $\varphi$ lies in the connected component of unity of the Shafarevich-Tate group and give a necessary condition for $X^\varphi$ to be bimeromorphic to a K\"ahler manifold.