Voisin's zero-cycle opposite-filtration conjecture

Let $X$ be a hyper-Kähler variety of dimension $2n$, and let $S_i\operatorname{CH}_0(X)$ denote the subgroup generated by classes of points whose rational-equivalence orbits have dimension at least $i$. Let $F'_{BB}$ denote the even part of the Bloch–Beilinson filtration. Voisin's conjecture. For every integer $i$ with $0\leq i\leq n$, the filtration $S_\bullet$ is opposite to $F'_{BB}$ in the sense that

$$S_i\operatorname{CH}_0(X)\cong \operatorname{CH}_0(X)/F'^{\,n-i+1}_{BB}\operatorname{CH}_0(X)=0.$$

The displayed equality is reproduced exactly from the source, although its final $=0$ makes the intended formulation require checking against the surrounding notation.