The constant cycle curve conjecture through the Beauville–Voisin class

Let $X$ be a complex K3 surface, and let $c_X\in\operatorname{CH}^2(X)$ be the distinguished Beauville–Voisin class. A constant cycle curve is a curve on $X$ whose points have a common class in the Chow group. Constant cycle curve conjecture. Every point $x\in X$ satisfying

$$[x]=c_X$$

is contained in a constant cycle curve. This geometric conjecture would provide a geometric version of the Bloch–Beilinson expectation; its general case is open.