On a conjecture of Matsushita

We prove a weak version of a conjecture of Matsushita saying that for a Lagrangian fibration on a hyper-Kaehler manifold $X$, the moduli map for the fibers is either generically of maximal rank or constant. Assuming the base is smooth and $b_2(X)-\rho (X)\ge 5$, we prove the conjecture for a very general deformation of $X$ with constant N'eron-Severi group.