You may have a look at Proposition 4.3.1 in our preprint https://arxiv.org/abs/1411.6354 (see page 47; $l^{c-1}$ denotes the localization of all effective motives by $c$-effective ones, whereas the only objects of "infinitely large weights" are zero ones). It easily yields that conjectures A and B in part II imply the "triangulated Hodge conjecture".
I am currently revising this text; in the next version a more detailed proof of a more general statement will be given.