Voisin's antisymmetrization conjecture for powers of K3 surfaces

Let $N\geq2$ be an integer and let $S$ be a K3 surface. Let $\operatorname{pr}:S^N\to S^{N-1}$ be projection onto the first $N-1$ factors, and let $p_{\wedge^N}$ denote the anti-symmetrization projector on $S^N$. It induces

$$\operatorname{pr}_*:(p_{\wedge^N})_*\operatorname{CH}_l(S^N)\longrightarrow(p_{\wedge^{N-1}})_*\operatorname{CH}_l(S^{N-1}).$$

Voisin's conjecture. For all $l<N$, the anti-symmetrization projector $p_{\wedge^{N+1}}$ acts as zero on $\ker(\operatorname{pr}_*)\otimes\operatorname{CH}_0(S)_{\mathrm{num}}$. The conjecture concerns the predicted vanishing of antisymmetric zero-cycle components on powers of a K3 surface; the supplied text does not indicate that it has been resolved.