Voisin's coisotropic-generation conjecture for hyper-Kähler varieties

Let $X$ be a projective hyper-Kähler manifold of dimension $2n$. For $i\leq n$, call a degree-$2i$ cohomology class coisotropic if it satisfies the coisotropic condition defined by the holomorphic symplectic form of $X$. Let $S_iX$ be the locus of points whose rational-equivalence orbit has dimension at least $i$. Voisin's conjecture. For any $i\leq n$, the space of coisotropic classes of degree $2i$ is generated over $\mathbb{Q}$ by classes of codimension-$i$ subvarieties $Z\subset X$ contained in $S_iX$. This refines the orbit-dimension conjecture by predicting that all coisotropic classes arise from these special subvarieties.