The elliptic Hall algebra realization of the positive Cartan subalgebra

Let $\mathcal E_{q_1,q_2,q_3}$ be the elliptic Hall algebra with $q_1q_2q_3=1$, and let $\mathcal E_{q^{-4},q^2,q^2}$ be its specialization at $(q^{-4},q^2,q^2)$. Let $\ddot{\mathrm{U}}_q^{0^+}(\mathfrak a_1)$ be the indicated closed positive Cartan subalgebra, and let its completion be understood in the topological sense. Elliptic Hall realization conjecture. $\ddot{\mathrm{U}}_q^{0^+}(\mathfrak a_1)$ is isomorphic to the completion of $\mathcal E_{q^{-4},q^2,q^2}$. The source has already constructed an algebra homomorphism in this setting and presents the isomorphism as a natural conjectural description; the claimed isomorphism is not established at this point.