Irreducible symplectic manifolds containing a Lagrangian torus

Recently, Beauville asked whether a Lagrangian torus in an irreducible symplectic manifold X is always the fiber of a Lagrangian fibration. In general, finding sufficient conditions for the existence of Lagrangian fibrations is one of the important questions in the field. In a joint project with Daniel Greb (Freiburg) and Sönke Rollenske (Bielefeld) we could answer Beauville's question in the affirmative in case that X is non-projective or of dimension 4. We also discuss certain reformulations of this question, hoping to promote a proof of the general case.