A flat morphism of schemes that is locally of finite presentation is universally open, for instance see The Stacks Project — Tag 01UA. I never encountered any reference to the inverse statement, and it seems too much to hope to be true. So, I would like to know if there are known examples of universally open morphisms of schemes that are not
- flat,
locally of finite presentation,flat or locally of finite presentation.
I am mainly interested in unramified such morphisms, yet any such example would be appreciated.