Satake equivalence for Hodge modules on affine Grassmannians

For a reductive group $G$ we equip the category of $G_\mathcal{O}$-equivariant polarizable pure Hodge modules on the affine Grassmannian $\mathrm{Gr}_G$ with a structure of neutral Tannakian category. We show that it is equivalent to a twisted tensor product of the category of representations of the Langlands dual group and the category of pure polarizable Hodge structures.