The Section Property for holomorphic symplectic varieties

Let $X$ be a smooth projective holomorphic symplectic variety. Write $\operatorname{CH}(X)$ for its Chow ring with rational coefficients and $\overline{\operatorname{CH}}(X)$ for the quotient by numerical equivalence. Section Property. The $\mathds{Q}$-algebra epimorphism

$$\operatorname{CH}(X)\twoheadrightarrow\overline{\operatorname{CH}}(X)$$

admits a section as $\mathds{Q}$-algebras whose image contains the Chern classes of $X$. This is a consequence expected from the splitting of the Bloch--Beilinson filtration; it is established for abelian varieties but remains open in the stated holomorphic symplectic setting.