The bounded-order constant cycle curve density conjecture

Let $X$ be a K3 surface. A constant cycle curve is a curve on $X$ whose points all have the same class in the appropriate Chow group; its order is the integer used to measure the associated generic-point class. Constant cycle curve density conjecture. There exists an integer $n>0$ such that the union

$$\bigcup C\subset X$$

of all constant cycle curves $C\subset X$ of order at most $n$ is dense. Claire Voisin proves this for generic complex K3 surfaces in the appendix, but the assertion for every K3 surface remains open.