We perform a key step towards the proof of Zvonkine’s conjectural r-ELSV formula that relates Hurwitz numbers with completed (r + 1)-cycles to the geometry of the moduli spaces of the r-spin structures on curves: we prove the quasi-polynomiality property prescribed by Zvonkine’s conjecture. Moreover, we propose an orbifold generalization of Zvonkine’s conjecture and prove the quasi-polynomiality property in this case as well. In addition to that, we study the (0, 1)- and (0, 2)-functions in this generalized case, and we show that these unstable cases are correctly reproduced by the spectral curve initial data.
