The noncommutative Hodge conjecture for small dg categories

Let $\mathcal{A}$ be a small dg category. Its Hodge classes are the subspace

$$\operatorname{Hodge}(\mathcal{A}):=\pi\left(j^{-1}\left(\operatorname{Ch}^{\operatorname{top}}\left(\operatorname{K}_{0}^{\operatorname{top}}(\mathcal{A})_{\mathbb{Q}}\right)\right)\right)\subset \operatorname{HH}_{0}(\mathcal{A}),$$

where $\operatorname{Ch}^{\operatorname{top}}$ is the topological Chern character, $j$ is the map from negative cyclic homology to periodic cyclic homology, and $\pi$ is the projection to Hochschild homology. Noncommutative Hodge conjecture. The Chern character

$$\operatorname{Ch}:\operatorname{K}_{0}(\mathcal{A})\longrightarrow \operatorname{HH}_{0}(\mathcal{A})$$

maps $\operatorname{K}_{0}(\mathcal{A})_{\mathbb{Q}}$ surjectively onto $\operatorname{Hodge}(\mathcal{A})$. This is intended as a noncommutative, non-weighted version of the rational Hodge conjecture; it recovers the classical conjecture for the dg category of perfect complexes on a smooth projective variety and agrees with the formulation for admissible subcategories described by Perry.