Voisin's standard conjecture on algebraic cycles supported on a closed subset

Let $X$ be a smooth complex projective variety and let $Y\subset X$ be a closed algebraic subset. Suppose $Z\subset X$ is a codimension-$k$ algebraic cycle whose class $[Z]\in H^{2k}(X,\mathbb Q)$ vanishes in $H^{2k}(X\setminus Y,\mathbb Q)$. Voisin's standard conjecture. There exists a codimension-$k$ cycle $Z'$ on $X$ with $\mathbb Q$-coefficients, supported on $Y$, such that $[Z']=[Z]$ in $H^{2k}(X,\mathbb Q)$. The source notes that Voisin observed this follows from the Lefschetz standard conjecture; it is therefore not treated as an established theorem here because the source presents it as a conjecture.