Noether–Lefschetz and tautological relation conjecture for moduli of K3 surfaces

For each integer $\ell\geq 1$, let $\mathcal M_{2\ell}$ denote the moduli space of polarized $K3$ surfaces of degree $2\ell$. Let ${\mathsf{NL}}^\star(\mathcal M_{2\ell})$ be the space of Noether–Lefschetz relations and let ${\mathsf{R}}^\star(\mathcal M_{2\ell})$ be the tautological ring of relations considered in the paper. Noether–Lefschetz relation conjecture. For all $\ell\geq 1$,

$${\mathsf{NL}}^\star(\mathcal M_{2\ell})={\mathsf{R}}^\star(\mathcal M_{2\ell}).$$

The conjecture is motivated by the abundance of relations obtained by localizing virtual classes of $\pi$-relative Quot schemes. The supplied text gives no resolution, so its status remains open.