Tate's pole-order conjecture for algebraic cycles

Let $X$ be a smooth projective geometrically connected variety over a finitely generated field $K$, let $d=\dim X$, and let $A^i(X)$ be the group of codimension-$i$ algebraic cycles modulo $\ell$-adic homological equivalence, with $N^i(X)$ the subgroup numerically equivalent to zero. For a smooth projective model $f:{\mathcal X}\to{\mathcal Y}$, let $\Phi_j(s)$ be the associated Euler products. Tate's pole-order conjecture. The dimension of $A^i(X)/N^i(X)$ equals the order of the poles of $\Phi_{2i}(s)$ at $s=\dim({\mathcal Y})+i$, and, by duality, of $\Phi_{2d-2i}(s)$ at $s=\dim({\mathcal X})-i$. This is a pole-order formulation of the relation between algebraic cycles and the relevant $L$-function; it remains open in general.