Generalized tautological conjecture for moduli of hyperkähler manifolds

Let $\mathcal{F}_h$ be the moduli space under consideration of hyperkähler manifolds, and let ${\rm NL}^\ast(\mathcal{F}_h)$ be the ring generated by Noether–Lefschetz cycles. Let $\mathrm{R}^\ast(\mathcal{F}_h)$ be the tautological ring generated by pushforwards of all $\kappa$-classes from the Noether–Lefschetz loci $\mathcal{F}_{\Sigma,h}$. Generalized tautological conjecture.

$${\rm NL}^\ast(\mathcal{F}_h)=\mathrm{R}^\ast(\mathcal{F}_h).$$

This generalizes the MOP conjecture from polarized K3 surfaces to hyperkähler manifolds. It asserts that all tautological Chow classes are generated by Noether–Lefschetz cycles, and the source presents it as an open proposal.