Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Let M be a holomorphically symplectic manifold, equipped with a Lagrangian fibration π:\; M → X. A degenerate twistor deformation (sometimes also called "a Tate-Shafarevich twist") is a family of holomorphically symplectic structures on M parametrized by H1,1(X). All members of this family are equipped with a holomorphic Lagrangian projection to X, and their fibers are isomorphic to the fibers of π. Assume that M is a compact hyperkahler manifold of maximal holonomy, and the Lagrangian projection π has no multiple fibers in codimension 1. We prove that M has a degenerate twistor deformation $M'$ such that the Lagrangian projection π:\; M' → X admits a meromorphic section.