Sections of Lagrangian fibrations on holomorphic symplectic manifolds

Let $M$ be a holomorphically symplectic manifold, equipped with a Lagrangian fibration $\pi:\; M \to X$. A degenerate twistor deformation (sometimes also called ``a Tate-Shafarevich twist'') is a family of holomorphically symplectic structures on $M$ parametrized by $H^{1,1}(X)$. All members of this family are equipped with a holomorphic Lagrangian projection to $X$, and their fibers are isomorphic to the fibers of $\pi$. Assume that $M$ is a compact hyperkahler manifold of maximal holonomy, and the general fiber of the Lagrangian projection $\pi$ is primitive (that is, not divisible) in integer homology. We also assume that $\pi$ has reduced fibers in codimension 1. Then $M$ has a degenerate twistor deformation $M'$ such that the Lagrangian projection $\pi:\; M' \to X$ admits a meromorphic section.