The genus-one Betti geometric Langlands conjecture

Let $E$ be a genus-one curve, let $\mathcal{H}$ be the universal affine Hecke category, and let

$$\mathcal{H}\otimes_{\mathcal{H}\otimes\mathcal{H}^{op}}\mathcal{H}\longrightarrow\mathcal{A}(E)$$

be the canonical map of Corollary~. Let $\operatorname{Loc}_{G^\vee}(E)$ denote the moduli space of $G^\vee$-local systems on $E$, and let $\operatorname{QC}^!_{\mathcal{N}}$ denote the category with the indicated nilpotent singular support condition. The genus-one Betti geometric Langlands conjecture. The map of Corollary~ is an equivalence. Hence

$$\mathcal{A}(E)\simeq\operatorname{QC}^!_{\mathcal{N}}(\operatorname{Loc}_{G^\vee}(E)).$$

This is the predicted Betti geometric Langlands equivalence for a genus-one curve, conditional in the source on the universal affine Hecke duality and related results.