Goulden–Jackson–Vakil parity conjecture for double Hurwitz polynomials

Let $\mathbf{x}=(x_1,\ldots,x_n)\in\mathbb Z^n$ satisfy $\sum_i x_i=0$, and let $H_g^r(\mathbf{x})$ be the double Hurwitz number, viewed piecewise polynomially on the chambers of the hyperplane $\sum_i x_i=0$. Goulden–Jackson–Vakil parity conjecture. On each chamber, the polynomial describing $H_g^r(\mathbf{x})$ has degree $4g-3+n$, has no nonzero terms of degree lower than $2g-3+n$, and is either even or odd, according to the parity of its leading coefficient. The theorem preceding the conjecture establishes piecewise polynomiality of degree $4g-3+n$; the asserted lower-degree bound and parity remain the conjectural content. Its status is not specified in the source.