Étale 1-motive realization conjecture in positive characteristic

Fix a perfect field $k$ of characteristic $p\geq 0$, and let $Sch/k$ denote the category of separated finite type $k$-schemes. Let $\mathscr{M}^1(k)$ be the category of free 1-motives over $k$, and let $T_\ell$ be their $\ell$-adic realizations for primes $\ell\ne p$. For the full subcategory $Sch_d/k\subset Sch/k$ of $d$-dimensional separated finite type $k$-schemes, the conjecture concerns functors

$$M^1(-),M_c^1(-):(Sch/k)^{op}\longrightarrow\mathscr{M}^1(k)$$

and

$$M^{2d-1}(-),M_c^{2d-1}(-):(Sch_d/k)^{op}\longrightarrow\mathscr{M}^1(k).$$

Étale 1-motive realization conjecture. These functors should satisfy, functorially in $X$, for every prime $\ell\ne p$,

$$T_\ell M^1(X)\cong H^1(X_{\overline{k}},\mathbb{Z}_\ell(1)),\qquad T_\ell M_c^1(X)\cong H_c^1(X_{\overline{k}},\mathbb{Z}_\ell(1)),$$

and, for $X\in Sch_d/k$,

$$T_\ell M^{2d-1}(X)\cong H^{2d-1}(X_{\overline{k}},\mathbb{Z}_\ell(d))/\mathrm{torsion},$$ $$T_\ell M_c^{2d-1}(X)\cong H_c^{2d-1}(X_{\overline{k}},\mathbb{Z}_\ell(d))/\mathrm{torsion}.$$

This is an $\ell$-adic analogue of the stated special case of Deligne's conjectures on 1-motives. The source restricts to perfect fields because it says that the conjecture is not clearly expected to hold over non-perfect fields; its resolution status is not specified.