Topology of projective Tate-Shafarevich twists

A Tate-Shafarevich twist $X^\phi\to B$ of a fibration $X\to B$ modifies it by a $1$-cocycle of flows of vector fields relative to the base, locally in the analytic topology. Sacc\`a conjectured that the total spaces of two projective Lagrangian fibrations related by such a twist are deformation-equivalent. Assuming that the class of the twist is torsion (which is often equivalent to the twist being realizable in the \'etale topology), we show that there is an isomorphism $H^\ast(X;\mathbb Q)\cong H^\ast(X^\phi;\mathbb Q)$ of graded vector spaces that respects (1) the Hodge structures and (2) the Hodge-Riemann pairing. Consequently, the rational Beauville-Bogomolov-Fujiki lattices of these two spaces are Hodge-similar. Assuming further that $B$ is smooth, and both the original fibration and its twist admit $C^\infty$-sections, we show Sacc\`a's conjecture using the theory of degenerate twistor deformations.