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?