Polynomial-growth conjecture for divisors and mapping-class-group orbits on fully ramified $Z_n$ curves

Let $X$ be the $Z_n$ curve associated to the family in the source, with parameters $p$ and $q$, and let $g$ be its genus. A divisor of degree $g$ is non-special when it is not special in the sense of the curve's divisor theory, and let $M$ denote the relevant mapping-class-group action on such divisors. Polynomial-growth conjecture. For sufficiently large $n$, the number of non-special divisors of degree $g$ on $X$ is a polynomial in $n$ of degree $q-1$. The number of $M$-orbits on these divisors is also a polynomial in $n$ of degree $q-1$. The conjecture predicts polynomial growth whose degree is determined by the parameter $q$; the examples discussed in the source exhibit this behavior for several values of $q$, but no general proof is given.