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