Section property conjecture for symplectic varieties

Let $X$ be a symplectic variety over an algebraically closed field. Let $\overline{\rm CH}^{*}(X)_{\mathbb{Q}}$ be the quotient of the rational Chow ring by numerical equivalence, and consider the algebra epimorphism

$${\rm CH}^{*}(X)_{\mathbb{Q}}\twoheadrightarrow\overline{\rm CH}^{*}(X)_{\mathbb{Q}}.$$

Section property conjecture. This algebra epimorphism admits a multiplicative section whose image contains all Chern classes of the tangent bundle of $X$.

This is the supersingular form of Beauville's splitting expectation: a multiplicative splitting should provide a distinguished subring of the Chow ring containing the tangent-bundle Chern classes. The source presents it as conjectural and does not prove it in general.