K3^[n]-type multiple-cover conjecture

Let $X$ be a variety of $K3^{[n]}$ type and let $\beta\in H_2(X,\mathbb{Z})$ be effective. For each divisor $k\mid\beta$, choose a variety $X_k$ of $K3^{[n]}$ type and a parallel transport lift $\varphi_k$ such that $\varphi_k(\beta/k)$ is primitive and has the same residue as $\beta/k$, up to sign. For tautological $\alpha$ and insertions $\gamma_1,\ldots,\gamma_N$, K3^[n]-type multiple-cover conjecture.

$$\left\langle\alpha;\gamma_1,\ldots,\gamma_N\right\rangle^X_{g,\beta}=\sum_{k\mid\beta}k^{3g-3+N-\deg(\alpha)}(-1)^{[\beta]+[\beta/k]}\left\langle\alpha;\varphi_k(\gamma_1),\ldots,\varphi_k(\gamma_N)\right\rangle^{X_k}_{g,\varphi_k(\beta/k)}.$$

This is the numerical multiple-cover form of the preceding conjecture and remains open.