Voisin's opposite-filtration conjecture for hyper-Kähler zero-cycles

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. Let $S_\bullet$ be the filtration on zero-cycles induced by the loci $S_iX$, and let $F'_{BB}$ denote the even levels of the Bloch–Beilinson filtration. Voisin's opposite-filtration conjecture. The filtration $S_\bullet$ is opposite to $F'_{BB}$ and thus provides a splitting of it. This would give a splitting of the Bloch–Beilinson filtration on $\operatorname{CH}_0(X)$ for general projective hyper-Kähler manifolds.