The rationality conjecture for locally w-Lefschetz 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 is locally $w$-Lefschetz if its specialization lies in the adelic span of the Lefschetz classes, and it is $w$-Lefschetz if its specialization is itself a Lefschetz class. Rationality conjecture. Every locally $w$-Lefschetz Hodge class on $A$ is $w$-Lefschetz. Equivalently, a Hodge class on $A$ fixed by the Lefschetz group $L(A_{0})$ specializes to a Lefschetz class on $A_{0}$. The paper identifies this with equality of two rational structures on the relevant invariant cohomology; its resolution is not supplied here.