Farkas–Pandharipande conjecture for the class ${\rm H}_g(Z-P)$

Let $m\geq 1$. For a smooth curve $(C,x_1,\ldots,x_{n+m})$ with markings, let $r$ be a positive integer and let $L$ satisfy

$$L^{\otimes r}\simeq \omega_C(-k_1x_1-\ldots-k_n x_n+p_1x_{n+1}+\ldots+p_mx_{n+m}).$$

Let $\overline{\mathcal{M}}_{g,Z-P}^{1/r}$ be the compactified moduli space of twisted $r$-spin structures, let $\pi:\overline{\mathcal{C}}_{g,Z-P}^{1/r}\to\overline{\mathcal{M}}_{g,Z-P}^{1/r}$ be its universal curve, let $\mathcal{L}$ be the universal line bundle, and let $\epsilon:\overline{\mathcal{M}}_{g,Z-P}^{1/r}\to\overline{\mathcal{M}}_{g,n+m}$ be the finite forgetful map. Define $c_g^r(Z-P)=c_g(R\pi_*\mathcal{L})$ and let $\widetilde{P}_{g,Z-P}$ be the asymptotic polynomial associated with $r\epsilon_*c_g^r(Z-P)$. The class ${\rm H}_g(Z-P)$ is the weighted sum of the classes of irreducible components in $A_g(\overline{\mathcal{M}}_{g,n+m})$. Farkas–Pandharipande conjecture. If $m\geq 1$, then

$${\rm H}_g(Z-P)=\widetilde{P}_{g,Z-P}(0)$$

in $A_g(\overline{\mathcal{M}}_{g,n+m})$. This conjecturally expresses the class defined by Farkas and Pandharipande in terms of the asymptotic polynomial of the $r$-spin construction; its status is not established by the supplied material.