Dolbeault homology vertex-algebra conjecture for affine Grassmannian bundles

Let $G$ be a group, with formal disc and loop groups $G(\mathcal{O})$ and $G(\mathcal{K})$, and let $\operatorname{Gr}_G$ be the affine Grassmannian. Let $\mathcal{V}$ be any vertex algebra without a Virasoro current, equipped with an action of $G(\mathcal{O})$. Form the associated bundle

$$G(\mathcal{K})\times_{G(\mathcal{O})}\mathcal{V}$$

on $\operatorname{Gr}_G$, denoted $\mathcal{V}_{\operatorname{Gr}_G}$. Dolbeault homology vertex-algebra conjecture. The Dolbeault homology of $\operatorname{Gr}_G$ with coefficients in $\mathcal{V}_{\operatorname{Gr}_G}$ has the structure of a vertex algebra.

This proposal is intended to construct the vacuum module, and the authors expect Beilinson–Drinfeld factorization machinery to establish the vertex-algebra structure, but they do not provide a rigorous proof here.