I was trying to unravel comments under the question #26269. It is apparently being claimed that if a proper morphism of finite presentation between schemes $X\rightarrow S$ is flat and has geometrically connected & reduced fibers then the natural map
$$
\mathcal{O}_S(S)\rightarrow \mathcal{O}_X(X)
$$
is an isomorphism. Is it true? By Grothendieck's coherence theorem the map is finite but I am not sure what to do next.