Let $\mathfrak{X}\rightarrow Spec(R)$ be a smooth family of smooth projective varieties over a local 1-dimensional ring of mixed characteristic. Suppose that there are non-empty subschemes (locally closed) $Y_0\subset Hilb_Q(\mathfrak{X}_{(0)}/Frac(R))$ and $y_m\subset Hilb_Q(\mathfrak{X}_{(m)}/k)$ (where $k=R/m$) defined by the same conditions (e.g. locally complete intersection and some cohomological conditions) in characteristic $0$ (generic fiber) and in characteristic $p$ (special fiber).

Is it always true that there is a subscheme $Z\subset Hilb_Q(\mathfrak{X}/Spec(R))$ such that $Z_{(0)}=Y_0$ and $Z_{(m)}=y_m$?