The abelian multiplicative geometric Langlands conjecture

Let $E$ be an elliptic curve, and let $\operatorname{Diff}_q(X)$ denote the category of $q$-difference modules on a variety or stack $X$ with automorphism $q$. The abelian multiplicative Langlands conjecture. For every $q \in \mathbb{CP}^1$, there is an equivalence of categories

$$\operatorname{Diff}_q(\operatorname{Bun}_{\mathrm{GL}(1)}(E)) \cong \operatorname{Coh}(q^{-1}\text{-Conn}_{\mathrm{GL}(1)}(E)).$$

This is the abelian elliptic case of the proposed multiplicative geometric Langlands correspondence. The source gives no proof or resolution, and later proposes a more sensitive two-parameter refinement.