The growth-rate conjecture for triangulations with at least six triangles at each internal vertex

Let $a_n$ be the number of triangulations of a fixed convex $n$-gon such that at least six triangles meet at each internal vertex. Let $b_n=\dim \operatorname{Inv}_{G_2}(V(\lambda_1)^{\otimes n})$, and let

$$B(x)=\sum_{n=0}^{\infty}b_nx^n.$$

The radius of convergence of the generating function $A(x)=1+x^2+\sum_n a_nx^n$ is at least $B(1/7)/7$. The growth-rate conjecture. If $a_n$ and $b_n$ are defined as above, then

$$\lim_{n\to\infty}\sqrt[n]{a_n}=\frac{7}{\sum_{n=0}^{\infty}b_n7^{-n}}=6.811\ldots.$$

Numerical evidence supports the assertion that the radius of convergence of $A(x)$ is exactly $B(1/7)/7$, equivalently that the displayed exponential growth rate holds. No proof or resolution is given in the source.