Brylinski–Deligne classification conjecture for factorizable line bundles

Let $X$ be a curve and $G$ a constant group scheme over $X$. Let $(K_2)_{\operatorname{Zar}}$ be the Zariski sheafification of the prestack assigning $K_2(A)$ to an affine scheme $\operatorname{Spec}(A)$ over $X$. Let $\operatorname{CExt}(G,(K_2)_{\operatorname{Zar}})$ be the Picard category of central extensions

$$1\to (K_2)_{\operatorname{Zar}}\to \widetilde{G}\to G\times X\to 1,$$

and let $\operatorname{FactPic}(\operatorname{Gr}_G)$ be the Picard category of factorizable line bundles on the affine Grassmannian. Brylinski–Deligne classification conjecture. The map

$$\operatorname{CExt}(G,(K_2)_{\operatorname{Zar}})\to \operatorname{FactPic}(\operatorname{Gr}_G)$$

is an isomorphism. The paper attributes this conjecture to earlier work and notes that it is known when $G=T$ is a torus, while the general case remains unresolved in the supplied context.