The equivalence conjecture for factorizable line bundles

Let $X$ be the smooth curve and $G$ the reductive group considered above, with maximal torus $T$. Write $B_{\operatorname{Zar}}(G)$ and $B_{\operatorname{Zar}}(T)$ for the corresponding Zariski classifying prestacks, $\bK_2$ for the second $K$-theory sheaf, and $\operatorname{FactLine}(\operatorname{Gr}_{G,\operatorname{Ran}})$ for the Picard groupoid of factorizable line bundles on the Ran affine Grassmannian. Then there is the constructed map of Picard groupoids

$$\operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^2_{\operatorname{Zar}}(\bK_2))\to \operatorname{FactLine}(\operatorname{Gr}_{G,\operatorname{Ran}}).$$

The equivalence conjecture. The map above is an isomorphism of Picard groupoids. This predicts that factorizable line bundles on the Ran affine Grassmannian are completely parameterized by the indicated based maps and hence by the corresponding $K$-theoretic data.