Cohomological tautological conjecture for moduli of hyperkähler manifolds

Let ${\rm NL}^\ast_{\rm hom}(\mathcal{F}_h)$ be the image of the Noether–Lefschetz ring in $H^\ast(\mathcal{F}_h,\mathbb{Q})$, and let $\mathrm{R}^\ast_{\rm hom}(\mathcal{F}_h)$ be the image of the tautological ring under the cycle class map. Cohomological tautological conjecture.

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

This is the cohomological weakening of the generalized tautological conjecture. The source proves the corresponding equality in the principal setting treated there, while the general conjecture for all hyperkähler moduli spaces remains open.