Does a conservativity conjecture (e.g. Conjecture 2.1 of <http://user.math.uzh.ch/ayoub/PDF-Files/Article-for-Steven.pdf>) imply the standard conjectures? Specifically I am confused with Beilinson's article arXiV:1006.1116 "Remarks on Grothendieck’s standard conjectures", where it seems to show that the conservativity of a realization functor implies the standard conjectures on algebraic cycles.



Does a conservativity conjecture imply the standard conjectures?
Here is a partial answer: In characteristic 0 it is known that conservativity + algebraicity *(edit: modulo rational equivalence)* of the Künneth projectors implies the remaining standard conjectures.


Indeed, if the Künneth projectors are algebraic, then one may use conservativity to show that the inverse of the Lefschetz operator is algebraic (i.e. Lefschetz standard conjecture). Once that is known, the hom=num standard conjecture also follows. For example, André's category of “motivated” motives would be equivalent to the categories of homological/numerical motives.


