On a conjecture of Matsushita

We prove that Lagrangian fibrations on projective hyper-Kähler 2n-folds with maximal Mumford-Tate group satisfy Matsushita's conjecture, namely the generic rank of the period map for the fibers of such a fibration is either 0 or maximal (i.e., n). We establish for this a universal property of the Kuga-Satake variety associated to a K3-type Hodge structure with maximal Mumford-Tate group.