Asymptotic ratio conjecture for prime webs

Let $k_n$ be the number of prime webs of $2n$ vertices.

Asymptotic ratio conjecture.

$$\lim_{n\rightarrow\infty} \frac{k_n}{k_{n-1}} = 3.829....$$

This conjectures an asymptotic growth ratio for the number of prime webs. The surrounding discussion relates prime webs to three-connected planar Eulerian triangulations, whose enumeration had been computed only up to $68$ vertices; no resolution is given here.