The motivic compatibility conjecture for factorizable line bundles and gerbes

Let $X$ be the smooth scheme considered above, let $G$ be the reductive group with maximal torus $T$, and let $\ell$ be the chosen prime. Consider the diagram

$$\begin{CD} \operatorname{Maps}(X\times B_{\operatorname{Zar}}(T),B^2_{\operatorname{Zar}}(\bK_2))/\ell @>>> \operatorname{Maps}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mu_\ell^{\otimes 2})) \\ @VVV @VVV \\ \operatorname{FactorLine}(\operatorname{Gr}_{G,\operatorname{Ran}})/\ell @>>> \operatorname{FactGerbe}_{\mu_\ell}(\operatorname{Gr}_{G,\operatorname{Ran}}). \end{CD}$$

The horizontal and vertical maps are the parameterization, Kummer, and gerbe maps constructed above. The motivic compatibility conjecture. The displayed diagram of equivalences commutes. This would identify the motivically constructed map with the Kummer-theoretic map on factorizable line bundles and gerbes.