A version of Hodge's conjecture due to Beilinson, expects that the Betti cycles class map $H_{\mathcal{M}}^i(X,\mathbb{Q}(j))\rightarrow hom_{MHS}(\mathbb{Q}(0),H^{i}(X,\mathbb{Q}(j) ))$ is surjective for smooth complex varieties $X$ that are defined over some number field (note that projectivity is not required). My question is whether it is possible to give a strengthening of the Hodge's conjecture (and similarly Tate's conjecture in char 0) which is stated as isomorphism of two cohomology theories? or as a quasi-isomorphism of sheaves belonging to two cohomology theories on the site of smooth schemes?
This is inspired from this paper which provides such a reformulation of the strong Tate's conjecture together with a conjecture of Beilinson on cycles over the finite fields. In general to me it seems more natural and maybe easier to prove two complexes of sheaves are quasi-isomorphic rather than prove that the map on the cohomologies is surjective.
I was thinking that maybe by combining the Hodge's conjecture and Voevodsky's nilpotence conjecture one could arrive to such a reformulation. The nilpotence conjecture expects the smash nilpotence adequate equivalence relation to coincide with the homological one. So it seems that there might be something similar to the motivic cohomology but designed for the smash nilpotence that can replace the motivic cohomology and turn the surjectivity to an isomorphism.
