Strict grading conjecture for birational motives of hyper-Kähler varieties

Let $X$ be a hyper-Kähler variety of dimension $2n$ and assume that its birational motive admits a co-multiplicative birational Chow–Künneth decomposition

$$\mathfrak{h}^{\circ}(X)=\mathfrak{h}^{\circ}_0(X)\oplus\mathfrak{h}^{\circ}_2(X)\oplus\cdots\oplus\mathfrak{h}^{\circ}_{2n}(X).$$

Strict grading conjecture. This unital grading is a strict grading. More strongly, the graded co-algebra morphism co-induced by the graded split surjection $\mathfrak{h}^{\circ}(X)\twoheadrightarrow\mathfrak{h}^{\circ}_2(X)$ is an isomorphism

$$\mathfrak{h}^{\circ}(X)\xrightarrow{\sim}\operatorname{Sym}^{\leq n}\mathfrak{h}^{\circ}_2(X).$$

This is described as a strengthening of the birational Chow–Künneth conjecture, motivated by the generalized Hodge conjecture and co-generation expectations; it remains open in general.