Voisin's vanishing conjecture for the Bloch–Beilinson filtration on constant-cycle classes

Let $X$ be a hyper-Kähler variety of dimension $2n$. For an integer $i$, let $C_i(X)\subset \operatorname{CH}_i(X)=\operatorname{CH}^{2n-i}(X)$ be the $\mathbb{Q}$-vector space generated by constant-cycle subvarieties of dimension $i$. Voisin's conjecture. The Bloch–Beilinson filtration satisfies

$$F_{BB}^{2n-2i+1}C_i(X)=0.$$

This condition is proposed so that a filtration assigning constant-cycle subvariety classes to the appropriate piece can be opposite to the Bloch–Beilinson filtration.