Voisin's Chern-class extension of the weak splitting conjecture

Let $X$ be a projective hyper-Kähler manifold, and let $C^*$ be the subalgebra of $CH^*(X)$ generated by divisors and Chern classes. Voisin's conjecture. The cycle class map is injective on $C^*$. This strengthens Beauville's weak splitting conjecture by including Chern classes among the generators; the source gives no general resolution.