Asymptotic enumeration conjecture for reduced historic trees

Let $m\geq 1$, and let $\rho_m$ be a positive constant associated with the dominant singularity of the generating function for reduced $((2m+1))$-historic trees. A reduced $((2m+1))$-historic tree has $n$ vertices and corresponds to a history of length $n+m$.

Asymptotic enumeration conjecture. For every $m\geq 1$, the number of reduced $((2m+1))$-historic trees with $n$ vertices is asymptotically equal to

$$n!\cdot\frac{(2m+1)!}{(m!)^2}n^m\rho_m^{-n-m-1}.$$

This conjecture follows from the expected dominant-singularity form of the solution to the higher-order differential equation governing the generating function. The statement predicts the asymptotic growth of reduced historic trees, but the supplied text does not establish the required singularity analysis or the existence and properties of $\rho_m$.