It seems to be well-known (see Is there an example of a variety over the complex numbers with no embedding into a smooth variety?) that a general finite type $S$-scheme does not embedd into a regular $S$-one even when $S$ is (the spectrum of) a field. Now, I am interested in the following setting: $X,Y$ are regular schemes of finite type over $S$; $S$ is separated excellent noetherian of finite Krull dimension (and one may assume that $X$ and $Y$ are proper over it). Then is it true that $X\times _S Y$ (or the corresponding reduced scheme) can be embedded into a regular scheme of finite type over $S$?

This is certainly true if $X,Y$ are quasi-projective over $S$ (so the question is not 'local'). On the other hand, if $S$ is smooth over a characteristic zero field $K$, then $X,Y$ are smooth over $K$, and $X\times_S Y\subset X\times_K Y$ (we can map the product to $S$ via the first component).

So, I wonder for what $S$ such an embedding always exists; I would be grateful for any hints or evidence!