Congruence conjecture for Hurwitz numbers in local Calabi–Yau geometry

Let $D_{d,g}$, $D^*_{d,g}$, and $D^{**}_{d,g}$ denote the weighted Hurwitz numbers for the relevant connected degree-$d$ covers of a genus-$g$ curve, and let $\widetilde N_d(g)$ be the local Gromov–Witten quantity appearing in the paper. Define

$$\Upsilon_{d,g}=216\widetilde N_d(g)$$

and, equivalently,

$$\Upsilon_{d,g}=(g-1)\left(27(g-1)D_{d,g}-27D^*_{d,g}-D^{**}_{d,g}\right).$$

Hurwitz-number congruence conjecture. If $d$ is not divisible by $4$, $6$, or $9$, then

$$\Upsilon_{d,g}\equiv 0\pmod{216}.$$

This congruence is a consequence predicted by the integrality of the local BPS invariants and translates the Gromov–Witten integrality expectation into arithmetic restrictions on the relevant Hurwitz numbers. The paper derives the formula leading to the congruence, but the stated divisibility prediction is conjectural.