The elliptic Hall algebra quotient conjecture for the punctured torus Homflypt skein

Let $\mathrm{Sk}(T^2-D^2)$ be the Homflypt skein algebra of the punctured torus, and let $\mathcal{E}_{\sigma,\bar\sigma}$ be the elliptic Hall algebra with generators $u_{\mathbf{x}}$ indexed by $\mathbf{x}\in\mathbb{Z}^2$. A simple closed curve has a homology class $\mathbf{x}\in\mathbb{Z}^2$. Elliptic Hall algebra quotient conjecture. There is a surjective algebra map

$$\mathrm{Sk}(T^2-D^2)\twoheadrightarrow\mathcal{E}_{\sigma,\bar\sigma}.$$

This map takes a simple closed curve of homology class $\mathbf{x}$ to the generator $u_{\mathbf{x}}\in\mathcal{E}_{\sigma,\bar\sigma}$ used by Schiffmann and Vasserot. The conjecture would identify the elliptic Hall algebra as a quotient of the larger punctured-torus Homflypt skein algebra and explain how its second parameter arises from the kernel of the map.