Toda's multiple cover conjecture for Euler-characteristic invariants on K3 surfaces

Let $S$ be a smooth projective K3 surface over $\mathbb{C}$, and set $X=S\times\mathbb{C}$. For an algebraic class $v=(r,\beta,n)\in H^0(S,\mathbb{Z})\oplus H^2(S,\mathbb{Z})\oplus H^4(S,\mathbb{Z})$, let $J(v)\in\mathbb{Q}$ be the Euler-characteristic DT-type invariant counting compactly supported $p^*\omega$-semistable sheaves on $X$ with Mukai vector $v$, where $p\colon X\to S$ is projection. Write $k\mid v$ when $v/k$ is an integral class, and let $(\,\ )$ be the Mukai pairing.

Toda's K3 multiple cover conjecture. We have

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

For primitive algebraic classes, the corresponding invariant is already given by the Euler characteristic of a Hilbert scheme of points on $S$; this conjecture extends that formula to nonprimitive classes. Its status is left open in the source, which presents it as a proposed multiple cover formula.