Let $f: X \to S$ be a proper flat morphism of schemes. Suppose that the geometric fibers of $f$ are connected and reduced. Does the condition $f_\ast(\mathcal O_X) = \mathcal O_S$ necessarily hold? (When $S = \mathop{\mathrm{Spec}}(k)$ is the spectrum of some field $k$, this statement is true by 0BUG. Conversely, if $f_\ast(\mathcal O_X) = \mathcal O_S$, then Stein factorization tells us the geometric fibers of $f$ are connected.)