The equivalence conjecture for the functor $\mathbb B^{\bullet}$

Let $k$ be the base field, let $F$ be the ambient field extension, and let $G$ be its field automorphism group. Let $\mathcal A dm$ denote the category of admissible representations of $G$ over $\mathbb Q$, and let $\mathbb B^{\bullet}$ be the functor from pure covariant motives over $k$ to graded semisimple admissible $G$-modules of finite length, with grading corresponding to powers of the motive $\mathbb L$. Equivalence conjecture. The functor $\mathbb B^{\bullet}$ is an equivalence of categories. This would identify pure covariant motives with the indicated graded representation-theoretic category; the source presents the assertion as expected, and gives no resolution.