Polynomial-degree conjecture for $\mathbb{Z}_2$ Hurwitz-Hodge integrals

Let $\vec{i}$ be a tuple of nonnegative integers, with $|\vec{i}|$ denoting the sum of its entries. Let $D_{(\vec{i}),2g+2}$ and $d_{(\vec{i}),2g+2}$ denote the corresponding $\mathbb{Z}_2$ Hurwitz-Hodge integrals as functions of $g$. Polynomial-degree conjecture. The integrals $D_{(\vec{i}),2g+2}$ and $d_{(\vec{i}),2g+2}$ are polynomials in $g$, and their degrees are precisely $2|\vec{i}|$. Theorem-level polynomiality is known with the weaker degree bound $|\vec{i}|^2+1$; the conjecture asserts the sharper exact degree $2|\vec{i}|$.