Let $\pi: X \to S$ be a morphism of schemes. I will say $\pi$ is "pseudoconnected" if $\mathcal{O}_S \to \pi_* \mathcal{O}_X$ is an isomorphism (this is not standard language).
If $\pi$ is proper with connected fibers, can we deduce that $\pi$ is pseudoconnected? I think this follows from Zariski's main theorem (the version that says a proper morphism is a pseudoconnected morphism followed by a finite morphism) but I am squeamish because Hartshorne doesn't explicitly say this (even for projective morphisms).
What if instead $\pi$ is proper with geometrically connected fibers?