Let's call a morphism of schemes a **strong immersion** if it is an open immersion followed by a closed immersion. This is no standard terminology. The following facts are well-known (see Stacks project, 19.24.3, 22.2.8, 22.2.9, 22.2.10):
 
- Every strong immersion is an immersion.
- Every quasicompact immersion is a strong immersion.
- Every immersion with a reduced domain is a strong immersion.
- There are immersions which are not strong.
 
Now my question is the following:
 
Let $X$ be an arbitrary scheme. Is the diagonal morphism $\Delta_X : X \to X \times X$ a strong immersion? According to the facts above, this is true when $X$ is reduced or when $X$ is quasi-separated.