Polynomial-growth conjecture for the number of curve diagrams

Let $g_{n,k}$ denote the number of curve diagrams of genus $n$ and complexity $k$, as in the paper. Let $n\geq 2$ be an integer. Polynomial-growth conjecture. There exist positive constants $\alpha_n$ and $\beta_n$ such that

$$\alpha_n k^{2(n-2)}\leq g_{n,k}\leq\beta_n k^{2(n-2)}$$

for every integer $k\geq 1$. This was already proved for $n=2$ and $n=3$; the conjecture predicts the precise polynomial order of growth for every fixed $n$ and is supported by the enumerated data.