Hard Lefschetz injectivity conjecture for Abel–Jacobi trivial Chow groups

Let $X$ be a smooth projective variety of dimension $n$, and let $L$ be an ample line bundle. Write $A^j_{AJ}(X)_{\Q}$ for the subgroup of Abel–Jacobi trivial codimension-$j$ cycles with rational coefficients. Hard Lefschetz injectivity conjecture. Intersection induces maps

$$\cdot L^{n-2j+2}\colon A^j_{AJ}(X)_{\Q}\to A^{n-j+2}(X)_{\Q}$$

that are injective for $2j-2\le n$. This is motivated by the Bloch–Beilinson conjectures and is related to a weak Lefschetz property for Chow groups; the source does not report a resolution.