The homological standard conjecture

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $\ell$ be a prime distinct from the characteristic of $k$. Numerical equivalence is the equivalence relation on algebraic cycles defined by equality of intersection numbers, while $\ell$-adic homological equivalence is defined by equality of their classes in $\ell$-adic étale cohomology. The homological standard conjecture. Numerical equivalence coincides with $\ell$-adic homological equivalence in the cohomology of $X$. This folklore assertion was stated by Tate and is also a consequence of the standard conjectures; it is open in general.