Hard Lefschetz conjecture for Lawson homology

Let $X$ be a smooth projective variety of dimension $n$, let $h$ be a hyperplane section, and let $[h]$ be its class in $L_{n-1}H_{2n-2}(X)_{\mathbb{Q}}$. The intersection operation induces

$$h^k\bullet:L_pH_{n+k}(X)_{\mathbb{Q}}\to L_{p-k}H_{n-k}(X)_{\mathbb{Q}},\qquad \alpha\mapsto [h]^k\bullet\alpha.$$

Hard Lefschetz conjecture for Lawson homology. The map $h^k\bullet$ is injective for $k\geq1$. This conjecture is presented as a Lawson-homology analogue of hard Lefschetz; the supplied text does not state that it has been resolved.