I am wondering wether the action of the Weyl group of a K3 surface $X$ is transitive on the sets of curves of fixed genus; in other words, given two curves $C,C'$ of genus $g$ on $X$, does there exists an element $\sigma$ of the Weyl group such that $\sigma C =C'$ ?

I am wondering wether the action of the Weyl group $W_X$ of a K3 surface $X$ is transitive on the sets of curves of fixed genus.

Suppose $W_X$ is non-trivial. Given two curves $C,C'$ of genus $g\geq2$ on $X$, does there exists an element $\sigma$ of the Weyl group such that $\sigma C =C'$ ?