  The Beauville-Voisin conjecture predicts the existence of a filtration on
projective hyper-K\"ahler manifolds opposite to the conjecture Bloch-Beilinson
filtration, called the Beauivlle-Voisin filtration. Voisin has introduced a
filtration on zero cycles of an arbitrary projective hyper-K\"ahler manifold.
On moduli space of stable objects of a projective K3 surface, there are other
candidates constructed by Shen-Yin-Zhao, Barros-Flapan-Marian-Silversmith and
more recently by Vial from different point of views. According to the work of
Vial, all of them are proved to be equivalent except Voisin's filtration. In
this paper, we show that Voisin's filtration is the same as the other
filtrations. As an application, we prove a conjecture in
Barros-Flapan-Marian-Silversmith's paper.
