General asymptotic formula for the enumeration polynomial

Prove that the enumeration polynomial $G(N,K)$ for rooted, ranked, unlabeled multifurcating tree shapes satisfies, for larger $K$, the hypothesized factorization and asymptotic coefficient formulas involving $a(K)=3K(K-1)/2+1$ and $b(K)=K(3K+7)/2$.

Background

The paper derives explicit polynomial expressions for $G(N,K)$ through $K=8$. These expressions exhibit a recurring leading-order form and a factorization by $(N-K)(N-K+1)$.

Based on the computed cases, the authors hypothesize that the same pattern persists for larger $K$, with specified formulas for the second asymptotic coefficients. No proof of this general pattern is provided in the manuscript, making it an explicit unresolved combinatorial question.