Tate–Beilinson conjecture on cycles and numerical equivalence over finite fields

Let $X$ be a smooth projective variety over $\mathbb{F}_q$ and let $n$ be an integer. Write $\hat G$ for the relevant Galois group, and let $H^{2n}_{cont}(\bar X,\mathbb{Q}_l(n))$ denote continuous $l$-adic cohomology of the base change $\bar X$. The cycle map is

$$CH^n(X)\otimes\mathbb{Q}_l\longrightarrow H^{2n}_{cont}(\bar X,\mathbb{Q}_l(n))^{\hat G}.$$

Tate–Beilinson conjecture. For every prime $l$: (1) the cycle map above is surjective; (2) the $\hat G$-module $H^{2n}_{cont}(\bar X,\mathbb{Q}_l(n))$ is semisimple at the eigenvalue $1$; and (3) rational and numerical equivalence agree with rational coefficients.

The source attributes the first two parts to Tate and the third to Beilinson. These conditions are presented as the classical conjecture related to the finite-generation and integral-comparison statements, but no resolution status is supplied.