I have a question regarding the equivariance in the Beilinson-Bernstein localization. Let $G$ be an simply connected algebraic group over a discrete valuation ring $R$ of mixed characteristic $(0,p)$ and $K$ a closed subgroup of $G$ with corresponding lie algebras $\mathfrak{g}, \mathfrak{k}$ and $X$ be the flag variety. One can define then the notions of $K$-equivariant $U(\mathfrak{g})$ modules and $K$-equivariant $D$-modules.

The Beilinson-Bernstein localization states that there is an equivalence between $K$-equivariant finitely generated $U(\mathfrak{g})$ modules with trivial central character and $K$-equivariant $D_X$ modules. This true when working over fields of characteristic 0 (One proof can be found in HTT theorem 11.5.3). The proof uses the crucial fact that $X$ is $D_X$- affine.

However, when working over a discrete valuation ring the flag variety is no longer $D_X$-affine. Can we still prove that the localization functor sends $K$-equivariant finitely generated $U(\mathfrak{g})$ modules with trivial central character to $K$-equivariant $D_X$ modules? My problem is that without $D$-affine condition I can not pass to the global sections. I am looking for any hints/references.

Remark: This is very much in the spirit of the question I asked here and got no response. https://math.stackexchange.com/questions/2498085/beilinson-bernstein-localization-equivariant-modules