In this MO answer of M. Bondarko, he says:

"the Hodge conjecture implies all the Grothendieck's standard conjectures over base fields of characteristic 0..."

and in Remarks on Grothendieck's standard conjectures Beilinson says:

"We show that Grothendieck’s standard conjectures (over a field of characteristic zero) follow from either of two other motivic conjectures, namely, that of existence of the motivic t-structure and (a weak version of) Suslin’s Lawson homology conjecture".

My question is: What about with standard conjectures in positive characteristic?