The multiplicative self-dual Chow--Künneth conjecture for holomorphic symplectic varieties

Let $X$ be a holomorphic symplectic variety, let $\operatorname{CH}(X)_{(0)}$ denote the degree-zero part of the Chow grading associated with a Chow--Künneth decomposition, and let $c_i(X)$ be its Chern classes. Multiplicative self-dual Chow--Künneth conjecture. The variety $X$ admits a multiplicative self-dual Chow--Künneth decomposition such that

$$c_i(X)\in \operatorname{CH}(X)_{(0)}$$

for every $i$. This is a motivic reformulation of Beauville's splitting principle and is known in several cases, including K3 surfaces, Hilbert schemes of points on K3 or abelian surfaces, and generalized Kummer varieties.