The Chow-ring intersection conjecture for Lagrangian fibrations

Let $X$ be a hyperkähler variety of dimension $4$ admitting a Lagrangian fibration, and let $A$ be a general fibre. Write $A^i(X):=CH^i(X)_{\mathbb{Q}}$ for the Chow groups with rational coefficients, and let $c_4(T_X)$ denote the fourth Chern class of the tangent bundle. Chow-ring intersection conjecture. One has

$$\operatorname{Im}\bigl(A^2(X)\xrightarrow{\cdot A}A^4(X)\bigr)=\mathbb{Q}[c_4(T_X)].$$

This predicts that intersecting codimension-two cycles with a general Lagrangian fibre produces precisely the one-dimensional subspace generated by the top Chern class. It is presented as a consequence of another conjecture together with the Bloch–Beilinson conjectures, and is not established in the source.