Quasi-polynomiality conjecture for monotone orbifold Hurwitz numbers

Let $a$ be a positive integer and let $mu_1,\dots,mu_n$ be positive integers. Define

$$C^{[a]}(\mu)=(a+1)^{\{\mu/a\}}\binom{\mu+\lfloor\mu/a\rfloor}{\lfloor\mu/a\rfloor}.$$

The quasi-polynomiality conjecture. The monotone orbifold Hurwitz numbers satisfy

$$\vec{H}_{g,n}^{[a]}(\mu_1,\ldots,\mu_n)=\prod_{i=1}^n C^{[a]}(\mu_i)\,Q_{g,n}^{[a]}(\mu_1,\ldots,\mu_n),$$

where $Q_{g,n}^{[a]}$ is a symmetric quasi-polynomial modulo $a$ of degree $3g-3+n$. This would extend the known polynomiality for monotone Hurwitz numbers in the case $a=1$ and is presented as a consequence of the topological recursion conjecture.