Let (X, L) be a polarized K3 surface of genus g and \(C_{en} \subset X\) be the curve of singular points of nodal elliptic curves in |L|. When (X, L) is generic of genus two, Huybrechts proved that the curve \(C_{en}\) is a constant cycle curve and conjectured that this remains true for higher genus cases. In this note, we show that the conjecture holds true for polarized K3 surfaces (X, L) lying in a locus of codimension one in the moduli space of polarized K3 surfaces of genus g for every \(g > 2\)
