Tate's numerical-cycles conjecture for varieties over finite fields

Let $V$ be a smooth projective variety over $\mathbb{F}_q$, let $r\geq 0$, and let $A^r(V)$ be the quotient of the space of algebraic cycles of codimension $r$ on $V$ by the subspace of cycles numerically equivalent to $0$. Let $\zeta(V,s)$ denote the zeta function of $V$. Tate conjecture. For all smooth projective varieties over $\mathbb{F}_q$ and $r\geq 0$, the dimension of $A^r(V)$ is equal to the order of the pole of $\zeta(V,s)$ at $s=r$. The paper explicitly says that the section assumes the full Tate conjecture, but the supplied material gives no resolution status for this formulation.