Tight closure/localization/geometric deformation over one dimensional domain/Fakt

Let  ${}\mathbb {Z} /(p)\subset D$  be a one-dimensional domain and  ${}D\subseteq R$  of finite type, and ${}I$ an ideal in ${}R$. Suppose that localization holds and that

$f\in I^{*}{\text{ holds in }}R\otimes _{D}Q(D)=R_{D^{*}}=R_{Q(D)}$

(${}S=D^{*}=D\setminus \{0\}$ is the multiplicative system). Then  ${}f\in I^{*}$  holds in ${}R\otimes _{D}\kappa ({\mathfrak {p}})$ for almost all ${}{\mathfrak {p}}$ in Spec ${}D$.