Tautologicality of reduced stable-map classes for lattice-polarized K3 surfaces

Let $\pi_\Lambda:\mathcal{X}_\Lambda\to\mathcal{M}_\Lambda$ be the universal lattice-polarized K3 surface, let $L\in\Lambda$ be an admissible class, and let

$$\epsilon^n:\overline{\mathsf{M}}_{g,n}(\pi_\Lambda,L)\to\mathcal{X}^n_\Lambda$$

be the evaluation map at the $n$ markings. The strict tautological ring $\mathsf{R}^\star(\mathcal{X}^n_\Lambda)$ is the subring generated by push-forwards from Noether–Lefschetz loci of products of relative diagonals, pull-backs of admissible divisor classes, pull-backs of $c_2(\mathcal{T}_{\pi_\Lambda})$, and pull-backs of strict tautological classes from the base. Tautologicality conjecture. The push-forward of the reduced virtual fundamental class lies in the strict tautological ring:

$$\epsilon_*^n\left[\overline{\mathsf{M}}_{g,n}(\pi_\Lambda,L)\right]^{\mathrm{red}}\in\mathsf{R}^\star(\mathcal{X}^n_\Lambda).$$

This predicts that reduced stable-map classes for lattice-polarized K3 surfaces are generated by the natural tautological geometric cycles. The required parts of this conjecture are proven by hand, so the statement is solved.