Shafarevich-Tate groups of holomorphic Lagrangian fibrations II

Let $X$ be a compact hyperkähler manifold with a Lagrangian fibration $π\colon X\to B$. A Shafarevich-Tate twist of $X$ is a holomorphic symplectic manifold with a Lagrangian fibration $π^φ\colon X^φ\to B$ which is isomorphic to $π$ locally over the base. In particular, $π^φ$ has the same fibers as