Perry's noncommutative integral Hodge conjecture

Let $\mathscr C$ be a geometric $\mathbf C$-linear category, meaning a semiorthogonal component of $\mathrm{D}_{\mathrm{perf}}(X)$ for a smooth, proper scheme $X$ over $\mathbf C$. Let $\operatorname{Hdg}(\mathscr C,\mathbf Z)$ denote the group of integral Hodge classes in $\mathrm{K}^{\mathrm{top}}_0(\mathscr C)$.

Perry's noncommutative integral Hodge conjecture. The homomorphism

$$\mathrm{K}_0(\mathscr C)\longrightarrow \operatorname{Hdg}(\mathscr C,\mathbf Z)$$

is surjective.

This extends the integral Hodge conjecture from smooth proper varieties to geometric linear categories and is formulated as a noncommutative Hodge-theoretic analogue. The source gives no resolution status for the conjecture.