Let $k$ be a field of characteristic zero and $DM_k$ be the derived category of rational Voevodsky motives. As I understand, there are conjectures which state that there is a $t$ structure on $DM_k$ such that the betti realization functor $DM_k \to Ch_{\mathbb Q}$ is exact and conservative (see Beilinson's Remarks on Grothendieck’s standard conjectures).

I am confused because (as Beilinson argues) this implies that betti realization yields a faithful functor from the heart of $DM_k$ to abelian groups. In particular, singular cohomology restricts to a fully faithful functor on the subcategory of pure chow motives under rational equivalence. This seems wrong to me, because homological equivalence of cycles is much stronger than rational equivalence of cycles.

For example over $\mathbb C$, the Hodge conjecture should only imply that the realization of the category of Chow motives is full, not faithful--since there are many cycles which are homologically equivalent but not rationally equivalent.

So from my perspective, it seems that these conjectures are wrong. Certainly, I am misunderstanding something-- what is it that makes the conjectures plausible?