Noncommutative Lefschetz standard conjecture for saturated dg-categories

Let $T$ be a saturated dg-category. Consider its algebraic classes in Hochschild homology, namely classes of the form $\alpha_E(\operatorname{Id}_E)$ for objects $E\in T$, and call a class antialgebraic if it pairs trivially with every algebraic class. Noncommutative Lefschetz standard conjecture. The trace pairing restricted to the algebraic classes is nondegenerate. In particular, any antialgebraic class is not algebraic. This is proposed as a dg-categorical analogue of the Lefschetz standard conjecture, extending the role of algebraic classes and trace pairings beyond varieties.