Let $k$ be an algebraically closed field of characteristic $p>0$ and $f:X \to Y$ be a quasi-projective morphism between noetherian $k$-schemes. Assume that $Y$ is regular and the geometric generic fiber of $f$ is generically non-reduced. Does there exist a noetherian scheme $Z$ and a dominant morphism $Z \to Y$, locally of finite type such that:

1) $Z$ is reduced and $\dim Z=\dim Y$

2) the geometric generic fiber of the composition $(Z \times_Y X)_{\mathrm{red}} \to Z \times_X Y \to Z$ is generically reduced.