Generalized Lefschetz standard conjecture for the coniveau filtration

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb C$, let $u\in H^2(X;\mathbb Z)$ be a hyperplane section class, and denote by $N^iH^{2i+k}(X)$ the coniveau filtration of coniveau $i$ and degree $2i+k$. For whole numbers $p,q,k$ satisfying

$$p+q=n-k,\qquad p\leq q,$$

consider the homomorphism given by cup product with $u^{q-p}$,

$$u^{q-p}_a:N^pH^{2p+k}(X)\longrightarrow N^qH^{2q+k}(X),\qquad \alpha\longmapsto \alpha\cdot u^{q-p}.$$

Generalized Lefschetz standard conjecture. For every such $p,q,k$, the homomorphism $u^{q-p}_a$ is an isomorphism. The assertion is presented as a strengthened or generalized form of the Lefschetz standard conjecture and is one of the assumptions underlying the paper's proposed method for constructing algebraic cycles. The supplied text does not indicate whether it has been proved or disproved.