Compatibility of Brylinski–Deligne and étale constructions

Let $X$ be a curve, let $G$ be a constant group scheme over $X$, and let $\ell$ be an integer prime to the characteristic. Set $A=\mu_\ell$, so that $A(1)\simeq \mu_\ell^{\otimes 2}$. Consider the map from central extensions by $(K_2)_{\operatorname{Zar}}$ to the étale mapping space and the map from factorizable line bundles to factorizable $\mu_\ell$-gerbes, together with the map from central extensions to factorizable line bundles. Brylinski–Deligne–étale compatibility conjecture. The resulting diagram commutes:

$$\begin{CD} \operatorname{CExt}(G,(K_2)_{\operatorname{Zar}}) @>>> \operatorname{Maps}_{\operatorname{Ptd}(\operatorname{PreStk}_{/X})}\left(B_{\operatorname{et}}(G)\times X,B^4_{\operatorname{et}}(\mu_\ell^{\otimes 2}\times X)\right) \\ @VVV @VVV \\ \operatorname{FactPic}(\operatorname{Gr}_G) @>>> \operatorname{FactGe}_{\mu_\ell}(\operatorname{Gr}_G). \end{CD}$$

Here the arrows are the constructions described immediately before the conjecture. This is stated as equivalent to a conjecture in the cited work; the supplied text gives no resolution.