Conjecture on intersections of Lagrangian subvarieties in the Chow ring

Let $X$ be a hyperkähler variety of dimension $2m$, and let $A^i(X):=CH^i(X)_{\mathbb{Q}}$ denote its codimension-$i$ Chow group with rational coefficients. Assume the Chow ring has pieces $A^i_{[j]}(X)$ indexed by the grade $j$. Let $Z\subset X$ be a Lagrangian subvariety, meaning an $m$-dimensional subvariety on whose regular part the symplectic form restricts to zero. Lagrangian intersection conjecture. The maps

$$A^j_{[j]}(X)\xrightarrow{\cdot Z}A^{m+j}(X)\longrightarrow A^{m+j}_{[j]}(X),$$

and

$$A^m_{[j]}(X)\xrightarrow{\cdot Z}A^{2m}(X)\longrightarrow A^{2m}_{[j]}(X)$$

are zero for all $j>0$, where the right arrows are projection to the piece $A^*_{[j]}(X)$. This is motivated by the vanishing of the action of a Lagrangian cycle on holomorphic forms and the expected Bloch–Beilinson-type grading of the Chow ring. The conjecture remains open in general.