The standard conjecture on supporting algebraic cycles on closed subsets

Let $X$ be a smooth complex algebraic variety, let $Y\subset X$ be a closed algebraic subset, and let $Z\subset X$ be an algebraic cycle of codimension $k$. Suppose its cohomology class satisfies

$$[Z]\in\operatorname{Ker}\bigl(H^{2k}(X,\mathbb{Q})\rightarrow H^{2k}(X\setminus Y,\mathbb{Q})\bigr).$$

Supporting-cycle conjecture. There exists a codimension-$k$ cycle $Z'$ on $X$ with rational coefficients, supported on $Y$, such that $[Z']=[Z]$ in $H^{2k}(X,\mathbb{Q})$. The paper relates this conjecture to the standard conjectures; it is known for codimension-$2$ cycles but open in general.