The factorization conjecture for the motivic parameterization map

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

$$\operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(T),B^2_{\operatorname{Zar}}(\bK_2))/\ell\to \operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mu_\ell^{\otimes 2})).$$

There is also the fully faithful map

$$\operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mathbb{Z}_\ell(2)))/\ell\hookrightarrow \operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mu_\ell^{\otimes 2})).$$

The factorization conjecture. The canonical map factors through this fully faithful map, and the first arrow in the factorization

$$\operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(T),B^2_{\operatorname{Zar}}(\bK_2))/\ell\to \operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mathbb{Z}_\ell(2)))/\ell\hookrightarrow \operatorname{Maps}_{\operatorname{based}}(X\times B_{\operatorname{Zar}}(G),B^4_{\operatorname{et}}(\mu_\ell^{\otimes 2}))$$

is an equivalence. Thus the canonical map is expected to arise from the $\ell$-adic motivic parameterization rather than merely from the Kummer construction.