Tate's strong conjecture on poles of zeta functions

Let $X$ be a smooth projective variety over a finitely generated field $F$, let ${\mathrm{Ch}}^r_{\hom}(X)$ be the quotient of the Chow group of codimension-$r$ cycles by $\ell$-adic homological equivalence, and, when $F$ is a number field, let $L(\operatorname{H}^{2r}(X)(r),s)$ be the incomplete $L$-function associated to the compatible system of Galois representations $\{\operatorname{H}^{2r}(X_{\overline F},\overline{\mathbb{Q}}_\ell(r))\}$. Tate's strong conjecture. Assume that $F$ is a number field. Then for any $1\le r\le \dim X$,

$$\operatorname{rank}{\mathrm{Ch}}_{\hom}^r(X)=-\operatorname{ord}_{s=1}L(\operatorname{H}^{2r}(X)(r),s).$$

This conjecture relates the rank of the group of homologically nontrivial algebraic cycles to the order of the pole of the corresponding $L$-function at $s=1$. The source states that the related independence-of-$\ell$ and injectivity assertion is known in characteristic zero, but gives no general resolution of this strong conjecture.