Jannsen's generalized Beilinson and semisimplicity conjecture

Let $k$ be a finite field, let $X$ be a variety over $k$, and let $i\in\mathbb{Z}$. Fix a prime $\ell\neq\operatorname{char}(k)$, write $\overline{X}$ for the base change to an algebraic closure, and let $\operatorname{Frob}_q$ denote geometric Frobenius, where $q=\lvert k\rvert$. The cycle class map is

$$\operatorname{CH}_i(X)\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\to H_{2i}(\overline{X},\mathbb{Q}_{\ell}(i))^{G_k}.$$

Generalized Beilinson and semisimplicity conjecture. The cycle class map induces an isomorphism, and the eigenvalue $1$ of $\operatorname{Frob}_q$ acting on $H_{2i}(\overline{X},\mathbb{Q}_{\ell}(i))$ is semisimple. In particular,

$$\dim_{\mathbb{Q}}\operatorname{CH}_i(X)=\operatorname*{ord}_{t=1}\chi(H_{2i}(\overline{X},\mathbb{Q}_{\ell}(i)),t).$$

This is the étale-homological extension of the preceding Tate and Beilinson-type conjectures from smooth varieties to arbitrary varieties over finite fields. The source attributes the formulation to Jannsen; no resolution status is supplied.