Voisin's injectivity conjecture for the tautological Chow ring of irreducible symplectic varieties

Let $X$ be an irreducible symplectic complex variety. Define $R(X)\subseteq \operatorname{CH}_{\mathbb Q}(X)$ to be the subalgebra generated by $\operatorname{CH}^1_{\mathbb Q}(X)$ together with the Chern classes $c_i(T_X)$ for $i\in\mathbb N$. Let $c_X$ denote the cycle class map.

Voisin's injectivity conjecture. The restriction

$$\left.c_X\right|_{R(X)}:R(X)\hookrightarrow H^*(X,\mathbb Q)$$

is injective.

This is a stronger version of Beauville's weak splitting conjecture, replacing the algebra generated by divisor classes with the algebra also generated by the Chern classes of the tangent bundle. Its status is not resolved in the supplied source.