The distinguished marking conjecture for holomorphic symplectic varieties

Let $X$ be a smooth projective holomorphic symplectic variety. A marking of $X$ is required to satisfy Condition $(\star)$, namely that the small diagonal is distinguished with respect to the product marking and that all Chern classes of $X$ are distinguished. Distinguished marking conjecture. Every such $X$ admits a marking satisfying $(\star)$. This condition is stronger than having motive of abelian type and is designed to behave well under motivic constructions such as products, blow-ups, and quotients; its general validity remains open.