The genus-one invariant conjecture for orbifolds

Let $\mathcal{X}$ be an orbifold, and let $II\mathcal{X}$ be its double inertia orbifold, whose points are triples $(x,g,h)$ with $x\in\mathcal{X}$ and commuting elements $g,h\in\operatorname{Stab}_\mathcal{X}(x)$. Let $\pi_2:II\mathcal{X}\to\mathcal{X}$ forget $g$ and $h$, and let $\langle\psi\rangle_{1,1,0}^{\mathcal{X}}$ and $\langle D\rangle_{1,1,0}^{\mathcal{X}}$ denote the relevant genus-one, degree-zero orbifold Gromov–Witten invariants. Genus-one invariant conjecture. One has

$$\langle\psi\rangle_{1,1,0}^{\mathcal{X}}=\frac{1}{24}\int_{II\mathcal{X}}c_{\operatorname{top}}(T_{II\mathcal{X}})=\frac{1}{24}\chi_{\operatorname{top}}(I\mathcal{X}),$$

and, for $D\in H^2(\mathcal{X})$,

$$\langle D\rangle_{1,1,0}^{\mathcal{X}}=\frac{1}{24}\int_{II\mathcal{X}}\pi_2^*(D)c_{\operatorname{top}-1}(T_{II\mathcal{X}}).$$

The conjecture comes from analyzing the moduli stack of genus-one, degree-zero orbifold stable maps with one non-stacky marked point. It holds for $\mathcal{X}=\mathcal{B}G$ by the cited work, but is otherwise to be addressed elsewhere.