The noncommutative Tate conjecture for saturated dg-categories

Let $k$ be a perfect field, let $l$ be a prime invertible in $k$, and let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $k$. Let $\operatorname{HK}_0(\operatorname{\mathcal{A}}\otimes_k\bar{k})_{\operatorname{\mathbb{Q}}_l}$ be the rationalized degree-zero periodic cyclic homology term and let $\pi_0(|r_l(\operatorname{\mathcal{A}}\otimes_k\bar{k})|)$ be the degree-zero $l$-adic realization, with its natural $\operatorname{Gal}(\bar{k}/k)$-action. Noncommutative Tate conjecture. The image of

$$\operatorname{ch}_l\otimes_k\bar{k}\colon \operatorname{HK}_0(\operatorname{\mathcal{A}}\otimes_k\bar{k})_{\operatorname{\mathbb{Q}}_l}\to \pi_0(|r_l(\operatorname{\mathcal{A}}\otimes_k\bar{k})|)$$

is the $\operatorname{Gal}(\bar{k}/k)$-invariant part of $\pi_0(|r_l(\operatorname{\mathcal{A}}\otimes_k\bar{k})|)$. It generalizes the classical Tate conjecture from varieties to saturated dg-categories; the source proposes it without resolving it.