Let $S$ be a base scheme, and $f : X \to S$ and $Y \to S$ finitely presented morphisms. Suppose that $f$ is faithfully flat and proper with connected reduced geometric fibers, and that $Y$ is affine over $S$. Is it true that every morphism $g : X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?