For a smooth projective variety $X$ over a field $k$ the Tate conjecture says that the cycle class maps $$CH^i(X)\otimes \mathbb{Q}_l \to H^{2i}(X,\mathbb{Q}_l(i))^{G_k}$$

are surjective. To my knowledge both the Chow groups and l-adic cohomology are representable in the motivic category by some ring spectra. Is there a way to interpret the cycle class map, and thus Tate's conjecture, in terms of these ring spectra? I haven't touched algebraic homotopy theory for a long time so this may be a naive question.