Voisin's injectivity conjecture for classes of constant-cycle subvarieties

Let $X$ be a hyper-Kähler variety of dimension $2n$. For each $i$, let $C_{2n-i}(X)\subset \operatorname{CH}^i(X)$ be the $\mathbb{Q}$-vector space generated by codimension-$i$ subvarieties contained in $S_iX$. Voisin's conjecture. For every $i$ with $0\leq i\leq n$, the cycle class map is injective on $C_{2n-i}(X)$. In particular, for $i=n$, it is injective on the subspace generated by classes of constant-cycle Lagrangian subvarieties. The paper notes evidence for this claim for the Fano variety of lines of a cubic fourfold.