The question is essentially in the title: $f\colon X \rightarrow Y$ is a monomorphism of schemes that is universally closed; does this imply that $f$ is a closed immersion? Any such $f$ is quasi-compact by https://stacks.math.columbia.edu/tag/04XU, so one may add this assumption if one wants.
In particular, is every integral monomorphism a closed immersion? (This would improve https://stacks.math.columbia.edu/tag/03BB)
If $Y$ is locally Noetherian, then the answer is positive and is Proposition 3.8 (i) in Ferrand's "Monomorphismes de schemas noetheriens". But is this perhaps true for any $Y$? Or are there known counterexamples? Of course, I do not want to assume that $f$ is of finite type ;)