Fourier–Mukai duality conjecture for cubic curves

Let $C:\operatorname{Spec} R\to \mathcal{M}_{cub}$ be a cubic curve, and let $\widehat{C}$ denote its formal completion, a formal group over $R$. Write $S^1_{\widehat{C}}=B\widehat{C}^{\vee}$. The category $\operatorname{QCoh}(C)$ admits an exotic symmetric monoidal structure $\star$ extending convolution on quasi-coherent sheaves over the smooth locus of $C$. Fourier–Mukai duality conjecture. There is a Fourier–Mukai symmetric monoidal equivalence

$$\operatorname{QCoh}(C)^\otimes\simeq\operatorname{QCoh}(C)^\star$$

which in particular induces an isomorphism

$$\operatorname{Aff}(C)\simeq S^1_{\widehat{C}}.$$

Such a theory would extend Fourier–Mukai duality from elliptic curves to arbitrary cubic curves over general base rings and would identify the Hochschild theory associated to the formal group with the mapping-space construction for the cubic. Its existence is left as a conjecture and is nontrivial even over algebraically closed fields of characteristic zero.