The integral geometric Satake equivalence in mixed characteristic

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1\mathrm{/}p]$ and $\mathcal{O}$ the its ring of integers. For a connected reductive group scheme $G$ over $\mathcal{O}$, we study the category $P_{L^{+}G}(G{r}_G,\mathrm{\Lambda })$ of $L^{+}G$-equivariant perverse sheaves in $\mathrm{\Lambda }$-coefficient on the affine Grassmannian $G{r}_G$ where $\mathrm{\Lambda }={\mathbb{Z}}_{\mathrm{\ell }}$ and ${\mathbb{F}}_{\mathrm{\ell }}$ and prove it is equivalent as a tensor category to the category of finitely generated $\mathrm{\Lambda }$-representations of the Langlands dual group of $G$.