Voisin's constant-cycle subvariety conjecture

Let $X$ be a hyper-Kähler manifold of dimension $2n$, and let $L,L'\subset X$ be $n$-dimensional constant-cycle subvarieties, meaning that all their points are rationally equivalent in $X$. Voisin's constant-cycle subvariety conjecture. If $[L]=[L']$ in $H^{2n}(X,\mathbb Q)$, then $L$ and $L'$ are rationally equivalent as algebraic cycles in $X$. This predicts that cohomological equivalence suffices for these special cycles; the source gives no resolution.