Let $f:X\to Y$ be a radiciel (=universally injective) morphism, where $X$ is a regular connected scheme. Can it be presented as the composition of a universal homeomorphism with an immersion? This seems to be equivalent to: the images of points of $X$ yield a subscheme of $Y$.

As in my previous questions, I am interested in excellent schemes of finite Krull dimension. If the answer to my question is 'no', then I would like to know whether there exists a 'short' characterization of the compositions of universal homeomorphisms with immersions.

Upd. As the comments show, the statement is wrong for trivial reasons if we don't assume that $X$ is connected.