The rationality conjecture for specializations of Hodge classes

Let $A$ be an abelian variety over an algebraic closure of $\mathbb{Q}$ with good reduction to an abelian variety $A_{0}$ over an algebraic closure of a finite field. A Hodge class on $A$ specializes to a cohomology class on $A_{0}$, and a Lefschetz class is a class generated by divisor classes and the Lefschetz operations. Rationality conjecture. The cup product of the specialization to $A_{0}$ of any Hodge class on $A$ with any Lefschetz class of complementary dimension lies in $\mathbb{Q}$. Equivalently, its $l$-adic component is a rational number independent of $l$. The conjecture follows for a given Hodge class when its specialization is algebraic, and is therefore implied by the Hodge conjecture for abelian varieties; it is known for CM abelian varieties with simple ordinary reduction and their powers.