Multiple-cover conjecture for sheaf-counting invariants on K3 surfaces

Let $S$ be a K3 surface, let $\widetilde{H}(S,\mathbb{Z})=\mathbb{Z}\oplus H^2(S,\mathbb{Z})\oplus\mathbb{Z}$ be its Mukai lattice, and let $v$ be an algebraic class in this lattice. Write $(v,v)$ for the Mukai pairing, let $k\mid v$ mean that $v/k$ is again an integral Mukai class, and let $\operatorname{Hilb}^m(S)$ denote the Hilbert scheme of $m$ points on $S$. Multiple-cover conjecture. The sheaf-counting invariant satisfies

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

The formula extends the known primitive-class calculation and is motivated by the multiple-cover behavior of related invariants. The source states that a complete computation for nonprimitive $v$ is not available, so the conjecture is presented as open.