Periodic-limit conjecture for normalized curve-diagram counts

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. Periodic-limit conjecture. There exists a positive integer $\rho_n$ such that, for every integer $\ell\in\{0,1,\ldots,\rho_n-1\}$, the sequence

$$\frac{g_{n,k\rho_n+\ell}}{k^{2(n-2)}}$$

has a positive limit $\lambda_{n,\ell}$ as $k\to+\infty$. This stronger conjecture is motivated by numerical patterns suggesting that the normalized counts split into convergent clusters according to a residue class of $k$; the paper mentions moduli $6$ for $n=4$ and $2$ for $n=5$, while the corresponding cases $n=2$ and $n=3$ exhibit related patterns.