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 ${\mathrm{G}\mathrm{r}}_G$ 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.