  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 $b_2(X)-\rho(X)\geq5$, we prove the conjecture for a very general
deformation of $X$ with constant N\'eron-Severi group.
