The quasi-coherent description conjecture for D-modules on the affine Grassmannian

Let $\mathcal{W}_n^{\mathrm{triv}}$ be the dg-stack defined from chains of local systems on $\mathcal{D}^*=\operatorname{Spec}(\mathbb{C}((z)))$, with the rank-$n$ local system identified with the trivial one, and let $\operatorname{Gr}_{\operatorname{GL}(n)}=\operatorname{GL}(n,\mathcal{K})/\operatorname{GL}(n,\mathcal{O})$ be the affine Grassmannian, where $\mathcal{O}=\mathbb{C}[[z]]$ and $\mathcal{K}=\mathbb{C}((z))$. The quasi-coherent description conjecture. The category $\operatorname{IndCoh}(\mathcal{W}_n^{\mathrm{triv}})$ is equivalent to the category $\operatorname{Dmod}(\operatorname{Gr}_{\operatorname{GL}(n)})$, and this equivalence respects the natural action of the tensor category $\operatorname{Rep}(\operatorname{GL}(n))$ on both sides. This conjecture was formulated as a corrected version of an earlier conjecture motivated by three-dimensional quantum field theory; the paper presents mathematical evidence, especially for $n=2$, but does not establish the equivalence in general.