Hard Lefschetz conjecture for Chow groups

Let $X$ be a smooth projective variety of dimension $n$, let $D$ be an ample divisor on $X$, and let $CH^p(X)$ denote the codimension-$p$ Chow group. If $n\geq 2p$, then multiplication by $D^{n-2p}$ defines a map

$$CH^p(X)\xrightarrow{\cdot D^{n-2p}}CH^{n-p}(X).$$

Hard Lefschetz conjecture for Chow groups. This map is injective. The conjecture is a Chow-theoretic analogue of the hard Lefschetz theorem on cohomology; surjectivity generally fails, and the statement is known in some cases, including abelian varieties over finite fields, where the map is an isomorphism.