Grothendieck's algebraicity conjecture for the Lefschetz operator

Let $X$ be a smooth projective variety of dimension $r$, let $H^*(X)$ be a Weil cohomology theory, and let $L$ be the Lefschetz operator. Let $\Lambda$ be the linear operator determined by the commutative diagrams relating $L$ and hard Lefschetz. A correspondence in $H^*(X\times X)$ acts on cohomology through the cycle map $\gamma^*_{X\times X}$. Grothendieck's algebraicity conjecture. The operator $\Lambda$ is algebraic: for every $Z\in \operatorname{CH}^i(X\times X)\otimes \mathbb{Q}$,

$$\Lambda(Z)=\gamma^*_{X\times X}(Z).$$

This is one of Grothendieck's standard conjectures, asserting that the inverse Lefschetz operation is induced by an algebraic correspondence; its status is not resolved in the source.