Hard Lefschetz conjecture for algebraic cycle cohomology

Let $X$ be a projective smooth variety over $F$ of dimension $n$ satisfying the paper's cohomological assumption, and let $L$ be an ample ${\mathbb R}$-divisor on $X$. Define $H^k(X)$ from the spaces of algebraic cycles modulo numerical equivalence, and let $L$ act by cup product.

Hard Lefschetz conjecture. For all $k$, the Lefschetz operator induces an isomorphism

$$L^k\colon H^{n-k}(X)\xrightarrow{\cong}H^{n+k}(X).$$

This is the hard Lefschetz input for the subsequent Hodge standard conjecture. In the paper it is used as an assumption when formulating positivity of the primitive pairings; its general status is not resolved here.