Voisin's polynomial decomposition conjecture for hyper-Kähler varieties

Let $X$ be a smooth projective hyper-Kähler $2n$-fold, let $h_X$ be a polarization, and let $Z_{\mathrm{lef}}\in\operatorname{CH}^2(X\times X)$ be a cycle such that its cohomological action $[Z_{\mathrm{lef}}]^*$ is the inverse of cup product with $h_X^{2n-2}$. Voisin's polynomial decomposition conjecture. There exist cycles $\gamma_i\in\operatorname{CH}^{2n-2i}(X)$ for $i=0,\ldots,n$, a divisor $D\subset X$, and a cycle $W\in\operatorname{CH}^{2n}(X\times X)$ supported on $D\times X$ such that

$$\Delta_X=\sum_{i=0}^n Z_{\mathrm{lef}}^i\cdot\operatorname{pr}_2^*(\gamma_i)+W\in\operatorname{CH}^{2n}(X\times X).$$

This conjecture would provide a polynomial decomposition of the diagonal for hyper-Kähler varieties and implies Voisin's effective zero-cycle conjecture for such varieties with respect to the Voisin filtration.