The K3 rational curves conjecture

Let $(X,H)$ be a projective K3 surface over an algebraically closed field, where $H$ is a polarization. The K3 rational curves conjecture. $X$ contains infinitely many integral rational curves $C$ linearly equivalent to some multiple of $H$. This conjecture is motivated by questions concerning hyperbolicity and rational points on K3 surfaces. It is known for very general K3 surfaces when the characteristic is not $2$ or $3$, but remains open in the stated generality.