Let all schemes below be excellent.
Let $X_0$ be a regular (not necessarily smooth, projective) variety over the generic fibre $S_0$ of a regular scheme $S$. As the answers to my question 
http://mathoverflow.net/questions/32083/for-a-morphism-f-from-a-regular-scheme-should-there-exist-an-open-subscheme-u-of
show, there does not have to exist an open $U\subset S$ such that $X_0$ possesses a regular model over $U$. Yet, should there always exist a pseudo-finite dominant morphism  $j:U\to S$ and some model $X$ of $X_0$ over $S$ such that $U$ and the reduced scheme associated to $X_U$ are regular? Can we assume that the morphism $U\to X$ is radical?