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 (that are weaker than Huber's assumptions) 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.

P.S. I would certainly be very glad if those who are interested in this question will read my paper. Yet I will try to outline the main idea of the argument here.

Firsly, it certainly suffices to verify that an effective geometric Voevodsky motif is $c$-effective (i.e., divisible by the cth power of the Lefschetz motif for some $c>0$) whenever we have the similar condition for its realization (yet one may actually miss this part of the reasoning). The main point is that it suffices to verify the natural analogue of the statement in question for complexes of Chow motives, whereas the localization of Voevodsky motives by $c$-effective ones "corresponds to" killing all morphisms of Chow motives that factor through c-effective ones. This is an easy consequence of the theory of weight structures (in particular, of the properties of weight complexes and localizations); you may also read my "Differential graded motives".