Strong piecewise polynomiality conjecture for double Hurwitz numbers

Fix $g$, $m$, and $n$, and let $P^{g}_{m,n}(\alpha_1,\dots,\alpha_m,\beta_1,\dots,\beta_n)=H^{g}_{\alpha,\beta}$ on the tuples of positive integers satisfying $\sum_i\alpha_i=\sum_j\beta_j$. The degree refers to the degrees of the polynomial pieces on the chambers of the associated piecewise-polynomial function. Strong piecewise polynomiality conjecture. $P^{g}_{m,n}$ is piecewise polynomial, with degrees between $2g-3+m+n$ and $4g-3+m+n$ inclusive. The lower bound is verified in the source in genus $0$ and when $m$ or $n$ equals $1$; the general assertion is left open there.