ℓ-adic integral Tate conjecture in codimension c

Let $X$ be a smooth projective variety over the algebraic closure $k$ of a finitely generated field, and let $\ell\neq\operatorname{char}(k)$ be a prime. The target of the cycle class map is the Tate submodule

$$\mathrm{H}^{2c}(X,\mathbf{Z}_{\ell}(c))^{\mathrm{Tate}}.$$

ℓ-adic integral Tate conjecture in codimension $c$. The cycle class map

$$\operatorname{CH}^c(X)\otimes\mathbf{Z}_{\ell}\to \mathrm{H}^{2c}(X,\mathbf{Z}_{\ell}(c))^{\mathrm{Tate}}$$

is surjective.

The paper discusses counterexamples to the integral Tate conjecture, so the general assertion is refuted.