MOP tautological conjecture for moduli of polarized K3 surfaces

Let $\mathcal{K}_g$ be the moduli space of primitively polarized K3 surfaces of genus $g$, and let ${\rm NL}^\ast(\mathcal{K}_g)$ be the subring of its Chow ring generated by Noether–Lefschetz cycles. Let $\mathrm{R}^\ast(\mathcal{K}_g)$ be the subring of ${\rm CH}^\ast_\mathbb{Q}(\mathcal{K}_g)$ generated by the images of the $\kappa$-classes on the Noether–Lefschetz loci $\mathcal{K}_\Lambda$. MOP conjecture.

$${\rm NL}^\ast(\mathcal{K}_g)=\mathrm{R}^\ast(\mathcal{K}_g).$$

This conjecture proposes that all tautological classes on the moduli space of polarized K3 surfaces are generated by Noether–Lefschetz cycles. Its cohomological version is proved in the paper, and the Chow-theoretic statement is essentially proved using Gromov–Witten theory, but the source does not state a complete resolution of the formulation given here.