Asymptotic growth conjecture for knotoid diagrams

Let $s\kappa_n$ be the number of $n$-crossing knotoid diagrams, let $sk_n$ be the number of $n$-crossing knotoid shadows, and let $\mu_K$ be the knotoid-shadow connective constant. Knotoid asymptotic-growth conjecture. There are constants $c'$ and $\gamma'$ such that

$$\frac{s\kappa_n}{2^n}=sk_n\mathop{\sim}_{n\to\infty}c'\mu_K^n\cdot n^{\gamma'-1},$$

with the additional belief that $\gamma'=\gamma$. This refines the exponential-growth assertion by predicting a power-law correction; the asymptotic form and the equality of exponents remain conjectural.