Multiple-cover conjecture for K3 sheaf-counting invariants on $\Gamma_0$

Let $S$ be a K3 surface, let $\Gamma_0$ be the lattice of Mukai classes used for the sheaf-counting invariants, and let $(v,v)$ denote the Mukai pairing. For $k\mid v$, assume $v/k$ is an integral class, and let $\operatorname{Hilb}^m(S)$ be the Hilbert scheme of $m$ points on $S$. Multiple-cover conjecture. For every $v\in\Gamma_0$,

$$J(v)=\sum_{k\geq 1,\ k\mid v}\frac{1}{k^2}\chi\left(\operatorname{Hilb}^{(v/k,v/k)/2+1}(S)\right).$$

This is the multiple-cover formula observed in the proof of the KKV theorem and is a restatement of the preceding multiple-cover conjecture with the more specific domain $v\in\Gamma_0$. It should therefore be merged with that conjecture rather than treated as a separate database entry.