General form conjecture for semiclassical moment generating functions

Let $g$ be a positive integer, let $\beta=1$ in the orthogonal case and $\beta=2$ in the unitary case, and let $\xi$ and $s$ be the variables used in the generating functions $T_{2g}^\beta$. Write $P_{2g}^\beta(\xi,s)$ for a polynomial in $\xi$ and $s$. General form conjecture. The generating functions have the form

$$T_{2g}^\beta=(\xi s)^\beta(1-s)^{-(2g+1)/2}(1-s+4\xi s)^{-(6g-1)/2}P_{2g}^\beta(\xi,s),$$

where $P_{2g}^\beta(\xi,s)$ is of order $2g-\beta$ in $\xi$ and of order $2(2g-\beta)$ in $s$. This proposed form is inferred from the cases $2g=1,2,3,4$ and predicts a uniform algebraic structure for the orthogonal and unitary moment generating functions beyond the computed orders.