Nori's conjecture on restrictions of cycles to subvarieties

Let $X$ be a smooth projective variety, let $i$ be an integer, and let $w\in\operatorname{CH}^i(X)$ be a cycle. Nori's conjecture. If $w|_T=0\in\operatorname{CH}_0(T)$ for every $i$-dimensional subvariety $T\subset X$, then

$$w=0\in\operatorname{CH}^i(X).$$

This predicts that a codimension-$i$ cycle is determined by its restrictions to all $i$-dimensional subvarieties; the paper uses it as an input for results on effective zero-cycles on abelian varieties.