Hecke-category realization conjecture for the theory $\calT[G]$

Let $G$ be a reductive group, let $\Gr_G$ be its affine Grassmannian, and let $D(\Gr_G)^{\operatorname{Hecke}}$ denote the derived category of Hecke eigen-modules, namely $D$-modules that are also right modules for the algebra $\calA_R$. Let $\calT[G]$ be the associated theory, with canonical Coulomb object $\calF_C$. Hecke-category realization conjecture. The category $\calC_C(\calT[G])$ is $D(\Gr_G)^{\operatorname{Hecke}}$, and $\calF_C=\calA_R$. This connects the Coulomb category of $\calT[G]$ with the Hecke-eigenmodule category on the affine Grassmannian; the source gives no resolution.