  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 observed 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 $g > 2$.
