The bicontinuous elliptic Hall algebra isomorphism for type a1

Let $\mathcal E_{q^{-4},q^2,q^2}$ be the specialized elliptic Hall algebra, let $\widehat{\mathcal E_{q^{-4},q^2,q^2}}$ denote its completion, and let $\ddot{\mathrm{U}}_q^{0^+}(\mathfrak a_1)$ be the positive Cartan subalgebra. Let $f$ be the previously constructed algebra homomorphism from the completed elliptic Hall algebra to this subalgebra. Bicontinuous isomorphism conjecture.

$$f:\widehat{\mathcal E_{q^{-4}, q^2,q^2}}\to\ddot{\mathrm{U}}_q^{0^+}(\mathfrak a_1)$$

is a bicontinuous $\mathbb F$-algebra isomorphism. This is the precise topological form of the preceding realization claim; the source postpones its proof, so it remains open in the stated context.