Orbifold Zvonkine ELSV formula for q-orbifold r-spin Hurwitz numbers

Let $h_{g,\mu_1,\dots,\mu_n}^{\circ,q,r}$ denote the $q$-orbifold $r$-spin Hurwitz numbers, let $\overline{\mathcal{M}}_{g,n}$ be the moduli space of stable curves, let $\psi_i$ be the cotangent-line classes, and let $\mathrm{C}_{g,n}$ be the relevant Chiodo class. For a real number $x$, write $\langle x\rangle$ for its fractional part and $\lfloor x\rfloor$ for its floor.

Orbifold Zvonkine formula. The $q$-orbifold $r$-spin Hurwitz numbers satisfy

$$h_{g,\mu_1,\dots,\mu_n}^{\circ,q,r}=\int_{\overline{\mathcal{M}}_{g,n}}\frac{\mathrm{C}_{g,n}\left(rq,q;qr-qr\left\langle\frac{\mu_1}{qr}\right\rangle,\dots,qr-qr\left\langle\frac{\mu_n}{qr}\right\rangle\right)}{\prod_{j=1}^n\left(1-\frac{\mu_i}{qr}\psi_i\right)}\\ \times r^{2g-2+n}(qr)^{\frac{(2g-2+n)q+\sum_{j=1}^n\mu_j}{qr}}\times\prod_{j=1}^n\frac{\left(\frac{\mu_j}{qr}\right)^{\left\lfloor\frac{\mu_j}{qr}\right\rfloor}}{\left\lfloor\frac{\mu_j}{qr}\right\rfloor!}.$$

The formula is proposed as a precise orbifold generalization of Zvonkine’s $r$-ELSV formula. The surrounding results establish quasi-polynomiality and the unstable initial data, while the formula itself is presented as a proposal rather than a proved theorem.