Nonemptiness conjecture for correspondences on very general polarized K3 surfaces

Let $(X,L)$ be a very general polarized K3 surface of genus $g\geq 3$, and let $Z_n(L)$ denote the correspondence locus introduced in the paper.

Nonemptiness conjecture. $Z_n(L)$ is not empty for any $g\leq 2n-2$.

The locus $Z_n(L)$ parametrizes pairs of points on curves in $|L|$ whose difference is $n$-torsion in the Jacobian. Brill–Noether theory gives the necessary inequality $g\leq 2n-2$, but the special ramification condition required here is not known to occur in general; the conjecture asserts that it does for every admissible pair $(g,n)$.