Elliptic character-sheaf Hochschild homology conjecture

Let $G$ be a complex reductive group, let $E$ be an elliptic curve, and let $\mathcal H_G$ be the affine Hecke category. Let $\mathcal Shv_{\mathcal N}(Bun_G(E))$ denote the category of elliptic character sheaves with nilpotent singular support. Elliptic character-sheaf Hochschild homology conjecture. The Hochschild homology category of the affine Hecke category $\mathcal H_G$ is equivalent to $\mathcal Shv_{\mathcal N}(Bun_G(E))$. The source explains that this is the automorphic statement to which the Betti Langlands conjecture for $E$ reduces, possibly in a unipotent version.