Conjecture on the asymptotic bound for analytic combinatorial curves

Let $A_k$ be the sequence appearing in the enumeration of analytic linear diagrams, let $a'_0$ be a constant independent of $k$, and let $\alpha$ be the exponential-growth parameter from the preceding theorem. For $k>0$, assume the bound

$$A_k\leq a'_0 k^{-\frac{3}{2}}\alpha^{-k}.$$

Asympture conjecture. The first terms of the sequence suggest that $a'_0=\alpha$ is sufficient; under this choice, the exponential-growth constant $\rho$ in the bound for the number of rooted analytic combinatorial curves on the sphere could satisfy $\rho<83$. More precisely, one may take

$$\rho=4\alpha^{-1}\exp\left(\sqrt{\frac{2a_0}{\sqrt{\pi}}}\right).$$

This conjectural estimate would improve the previously established bound $\rho\leq96e^{1/3}<134$. Further information about the distributions of $s$ and $k$ as the number of edges tends to infinity could yield an even smaller value of $\rho$.