Grothendieck–Hodge conjecture for the weight-graded cohomology of singular schemes

Let $X$ be a proper integral scheme over $\mathbb C$, and let $\nu^{i,l}$ be the natural map

$$\nu^{i,l}:H^i((N^lH^t)^\bullet)\to \operatorname{gr}^W_tH^{t+i}(X),$$

where $(N^lH^t)^\bullet$ is induced by the coniveau filtration on the components of a smooth proper hypercovering. Its image lies in $\operatorname{gr}^W_tH^{t+i}(X,\mathbb Q)\cap F^l$. Grothendieck–Hodge conjecture. The image of $\nu^{i,l}$ is the largest subspace of $\operatorname{gr}^W_tH^{t+i}(X,\mathbb Q)\cap F^l$ that is a sub-Hodge structure of $\operatorname{gr}^W_tH^{t+i}(X)$. This extends the coniveau formulation of the Grothendieck–Hodge conjecture from smooth schemes to weight-graded cohomology of singular schemes; it is presented as conjectural.