The Catalan-polynomial structure conjecture for the moment generating function of random graphs

Let $M_{\mu^c}(z)$ be the ordinary moment generating function of $\mu^c$, and let $C(z)$ denote the Catalan generating function. For a formal power series in $c^{-1}$, write $[c^{-i}]$ for the coefficient of $c^{-i}$. Define

$$V_i(z):=[c^{-i}]M_{\mu^c}\left(\sqrt{\frac{z}{P(1/c)}}\right),\qquad i\geq 0.$$

Catalan-polynomial structure conjecture. There exists a unique power series $P(x)$ with non-negative integer coefficients such that every $V_i(z)$ is the product of $C(z)$ and a polynomial in $zC(z)^2$.

This conjecture predicts a uniform algebraic structure for all coefficients in the large-$c$ expansion of the moment generating function. The preceding asymptotic theorem motivates the normalization by $P(1/c)$, while the conjecture leaves the existence and uniqueness of the required power series, as well as the asserted factorization for every $i$, open.