A Tate-Shafarevich twist of a (proper) fibration modifies it by a 1-cocycle of automorphisms given by flows of (holomorphic) vector fields relative to the base, locally in the analytic topology. In general, the total space of a twist does not even have to be homeomorphic to that of the original fibration. Nevertheless, it was conjectured by Saccà that if one started with a Lagrangian fibration of an irreducible hyper-Kähler variety, then the total space of the resulting twist should always be deformation-equivalent to that of the original fibration, provided that it is also algebraic. I will introduce evidence towards this conjecture, including coincidences of certain cohomological invariants, as well as a proof under further topological constraints.