Huybrechts' conjecture on curves of elliptic nodes on polarized K3 surfaces

Let $(X,L)$ be a polarized K3 surface of genus $g\geqslant 3$. Let $C_{en}$ denote the curve of elliptic nodes of $(X,L)$, namely the curve swept out by the singular points of the nodal elliptic curves in $|L|$. A curve on a K3 surface is a constant cycle curve if all of its points represent the same class in $\operatorname{CH}_0(X)$. Huybrechts' conjecture. If $C_{en}$ is nonempty, then $C_{en}$ is a constant cycle curve. The genus-two case was observed by Huybrechts, while this conjecture concerns the higher-genus cases $g\geqslant 3$; the paper proves it for polarized K3 surfaces lying in a codimension-one locus in the moduli space for every $g>2$, leaving the general case open.