Motivic splitting property for holomorphic symplectic varieties

Let $X$ be a holomorphic symplectic variety of dimension $2n$. Then the motivic splitting property. there is a canonical self-dual multiplicative Chow–Künneth decomposition

$$\mathfrak h(X)=\bigoplus_{i=0}^{4n}\mathfrak h^i(X),$$

of Bloch–Beilinson–Murre type: for every $i,j\in\mathbb N$, $\operatorname{CH}^{i}(\mathfrak h^{j}(X))=0$ if $j<i$ or $j>2i$, and realization induces an injective map

$$\operatorname{Hom}_{\operatorname{CHM}}\left(\mathbf 1(-i),\mathfrak h^{2i}(X)\right)\to\operatorname{Hom}_{\mathbb Q\text{-HS}}\left(\mathbb Q(-i),H^{2i}(X)\right).$$

This motivic formulation refines Beauville's splitting property and would imply the corresponding Chow-ring statement. The supplied source gives no general proof.