Voevodsky's smash-nilpotence conjecture

Let $X$ be a smooth projective variety over a field $k$, and let a cycle on $X$ be smash-nilpotent if some positive external power of it is rationally equivalent to zero on a corresponding power of $X$. Voevodsky's conjecture. Any cycle numerically equivalent to $0$ is smash-nilpotent. This is known when numerical and algebraic equivalence agree, including cycles of dimension $0$ and codimension $1$, and in several further cases such as skew cycles on abelian varieties and $1$-cycles on products of curves; the general assertion remains open.