Milne's rationality conjecture for absolute Hodge classes

Let $A$ be an abelian variety over $\mathbb{Q}^{\mathrm{al}}$ with good reduction $A_0$ over $\mathbb{F}^{\mathrm{al}}$, let $d=\dim(A)$, and let $\gamma$ be an absolute Hodge class of codimension $i$ on $A$. Let $\gamma_l\in H^{2i}(A_0,\mathbb{Q}_l(i))$ for $l\ne p$ and $\gamma_p\in H_{\mathrm{crys}}^{2i}(A_0/W)(i)_{\mathbb{Q}}$ be its specializations. For divisors $D_1,\ldots,D_{d-i}$ on $A_0$, write $\delta_m(l)$ for their $l$-cohomology classes. Rationality conjecture. The intersection number

which a priori lies in $\mathbb{Q}_l$ or $(\mathbb{Q}^{\mathrm{al}})_w$, is a rational number independent of $l$. This conjecture concerns compatibility of realizations of absolute Hodge classes under reduction and is used in constructing motivic categories with rational Tate classes. Its general status is not resolved in the supplied text.