  A conjecture of Voisin states that two points on a smooth projective complex
variety whose algebra of holomorphic forms is generated in degree 2 are
rationally equivalent to each other if and only if their difference lies in the
third step of the Bloch-Beilinson filtration. In this note, we formulate a
generalization that allows for rational equivalence of effective zero-cycles of
higher degree, at the expense of looking deeper in the Bloch-Beilinson
filtration. In the first half, we provide evidence in support of this
conjecture in the case of abelian varieties and projective hyper-K\"ahler
manifolds. Notably, we give explicit criteria for rational equivalence of
effective zero-cycles on moduli spaces of semistable sheaves on K3 surfaces,
generalizing that of Marian-Zhao. In the second half, in an effort to explain
our main conjecture, we formulate a second conjecture predicting when the
diagonal of a smooth projective variety belongs to a subalgebra of the ring of
correspondences generated in low degree.
