The quantum loop algebra conjecture for observables on the cylinder

Let $\mathfrak{g}$ be the Lie algebra used to define the gauge theory, let $\operatorname{Obs}^q({\mathbb{C}}^\times \times {\mathbb{R}}^2)$ be the resulting quantized $E_2$-algebra of observables, and let $U_\hbar(\mathfrak{g}\{z,z^{-1}\})$ denote the quantum loop algebra. Quantum loop algebra conjecture. The Hopf algebra Koszul dual to the $E_2$-algebra $\operatorname{Obs}^q({\mathbb{C}}^\times \times {\mathbb{R}}^2)$ is dual to the quantum loop algebra $U_\hbar(\mathfrak{g}\{z,z^{-1}\})$. There are subtle issues concerning which completion of $\mathfrak{g}[z,z^{-1}]$ gives precisely the Koszul dual associated to the cylinder, so the precise formulation may require refinement.