Call a morphism $\pi:\ X \to Y$ of schemes $\mathcal{O}$-connected if the natural map $\mathcal{O}_Y \rightarrow \pi_\ast \mathcal{O}_X$ is an isomorphism.

Suppose that $\pi: X \rightarrow Y$ is a morphism of $k$-schemes, where $k$ is an algebraically closed field. Suppose also that the following hypotheses hold.

- $X$ and $Y$ are reduced.
- $\pi$ is projective.
- All (geometric) fibers of $\pi$ are reduced, nonempty, and connected.

Does it follow that $\pi$ is $\mathcal{O}$-connected?

I am aware that it would be sufficient to assume that $\pi$ is flat (for example, by Exercise 28.1.H in Ravi Vakil's *The Rising Sea*). However, I'm hoping to find a criterion that would apply to certain non-flat situations, such as blow-ups. It seems that it would also be sufficient to assume that $Y$ is normal (in which case one doesn't need the hypothesis that the fibers are reduced), as is discussed at the following question: When will the pushforward of a structure sheaf still be a structure sheaf?

I'm wondering if it is possible to avoid the hypothesis that $Y$ is normal, in the situation described above (in particular, in cases where the fibers are reduced). I would like to find a good reference if so.