The Chow-ring multiplicativity conjecture for the Lefschetz eigenspace decomposition

Let $X$ be a hyperkähler variety of $K3^{[2]}$-type endowed with a lift $L\in {\rm CH}^2(X\times X)$ of $\mathfrak{B}$ satisfying the stated relations, and suppose that $L$ additionally satisfies the relation defining $L^2$. Let $H$ be the induced operator and write $\Lambda^i_\lambda\subseteq {\rm CH}^i(X)$ for the eigenspace of eigenvalue $\lambda$ of $H_\ast$, with ${\rm CH}^i(X)_s:=\Lambda^i_{2i-4-s}$. For all occurring $s,t\in\mathbb Z$, intersection product is conjectured to define a map

$${\rm CH}^i(X)_s\times {\rm CH}^j(X)_t\longrightarrow {\rm CH}^{i+j}(X)_{s+t}.$$

Multiplicativity conjecture. The intersection product preserves the eigenspace grading: products of classes in ${\rm CH}^i(X)_s$ and ${\rm CH}^j(X)_t$ lie in ${\rm CH}^{i+j}(X)_{s+t}$.

This would make the Lefschetz eigenspace decomposition a multiplicative Chow decomposition, analogous to Beauville's decomposition for abelian varieties. The theorem preceding the conjecture establishes the decomposition and injectivity of the cycle class map on most degree-zero pieces, but multiplicativity remains conjectural.