The rationality conjecture for CM abelian varieties

Let $A$ be a CM abelian variety over $\mathbb{Q}^{\mathrm{al}}$, with good reduction $A_0$ over $\mathbb{F}$. A Hodge class on $A$ is an element of $H_{\mathbb{A}}^{2*}(A)(*)$, and its specialization is a class on $A_0$; a Lefschetz class on $A_0$ is generated by divisor classes. Rationality conjecture. The product of the specialization to $A_0$ of any Hodge class on $A$ with any Lefschetz class on $A_0$ of complementary dimension lies in $\mathbb{Q}$. Equivalently, the finite-component value is a rational number independent of $\ell$. The conjecture would support the construction of a good theory of rational Tate classes; the source notes it holds in particular when the specialized Hodge class is algebraic and is implied by the Hodge conjecture for CM abelian varieties.