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 For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular 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?