Generalized Grothendieck–Hodge conjecture

Let $X$ be a smooth complex projective variety, and let $L\subset H^k(X,\mathbb{Q})$ be a sub-Hodge structure of coniveau $c$. The coniveau of a rational Hodge structure of weight $k$ is the smallest integer $c$ such that $L^{k-c,c}\not=0$. Generalized Grothendieck–Hodge conjecture. There exists a closed algebraic subset $Z\subset X$ of codimension $c$ such that $L$ vanishes under the restriction map

$$H^k(X,\mathbb{Q})\rightarrow H^k(U,\mathbb{Q}),$$

where $U:=X\setminus Z$. This conjecture relates the coniveau of a Hodge structure arising from geometry to support in codimension $c$; its status is not resolved in the supplied source.