Proved only in codimension one, by the Lefschetz theorem on (1,1) classes. Higher codimension cases, including named examples such as the diagonal cycle's Kunneth components, remain open, and the natural integral version of the conjecture is known to be false.
What would resolve this A general proof for every codimension, or an explicit Hodge class shown not to be a combination of algebraic cycle classes.
OpenAlgebraic geometryClay Mathematics Institute