The noncommutative Hodge conjecture for saturated dg-categories

Let $\operatorname{\mathcal{A}}$ be a saturated dg-category over $\operatorname{\mathbb{C}}$, and let $\operatorname{Hdg}(\operatorname{\mathcal{A}})_{\operatorname{\mathbb{Q}}}$ denote its rational noncommutative Hodge classes. Noncommutative Hodge conjecture. The cycle map

$$K_0(\operatorname{\mathcal{A}})_{\operatorname{\mathbb{Q}}}\to \operatorname{Hdg}(\operatorname{\mathcal{A}})_{\operatorname{\mathbb{Q}}}$$

is surjective. This extends the classical Hodge conjecture to saturated dg-categories; the source proposes the statement and does not establish it in general.