Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K  3 surface. Beauville and Voisin singled out a 0-cycle _cX$c_X$ on X of degree 1 and Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X   is a multiple of _cX$c_X$ if certain hypotheses hold. We believe that the following generalization of Huybrechts? result holds. Let M be a moduli space of stable pure sheaves on X with fixed cohomological Chern character: the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) depends only on the dimension of M. We will prove that the above statement holds under some additional assumptions on the Chern character.