The cohomological singular Hodge conjecture

Let $X$ be a projective variety such that the dimension of its singular locus is at most $p-1$. The cycle class map is

$$\mathrm{cl}_p:A^p(X)\otimes\mathbb{Q}\to \operatorname{Gr}^W_{2p}H^{2p}(X,\mathbb{Q}),$$

where $A^p(X)$ is the operational Chow group. Define

$$H^{2p}_{\mathrm{Hdg}}(X):=\operatorname{Gr}^W_{2p}H^{2p}(X,\mathbb{Q})\cap F^p\operatorname{Gr}^W_{2p}H^{2p}(X,\mathbb{C}).$$

The cohomological singular Hodge conjecture. The image of $\mathrm{cl}_p$ coincides with $H^{2p}_{\mathrm{Hdg}}(X)$. This extends the classical Hodge conjecture to singular varieties using operational Chow groups and the weight filtration; its general status is not specified in the source.