Polynomial structure conjecture for Masur–Veech volumes

Let $MV_{g,n}$ denote the Masur–Veech volume, and set

$$\gamma_k=\frac{1}{4^k}\binom{2k}{k}.$$

For each $g\geq 0$, consider polynomials $p_g,q_g\in\mathbb{Q}[n]$. Polynomial structure conjecture. For any $g\geq 0$, there exist such polynomials with

$$\deg p_g=\begin{cases}\lfloor(g-1)/2\rfloor,&g>0,\\-\infty,&g=0,\end{cases} \qquad\text{and}\qquad \deg q_g=\lfloor g/2\rfloor,$$

and, for any $n\geq 0$,

$$\frac{MV_{g,n}}{\pi^{6g-6+2n}}=2^n\frac{(2g-3+n)!(4g-4+n)!}{(6g-7+2n)!}\left(p_g(n)+\gamma_{2g-3+n}q_g(n)\right).$$

This predicts a uniform polynomial description of the volumes in every genus and number of marked points; the supplied text gives no resolution of the conjecture.