Tube-polymer link-statistics conjecture

Let $\mathbb T=\mathbb T_{M_1,M_2}$ be a lattice tube, and let $L$ be a non-split link embeddable in $\mathbb T$. Let $p_{\mathbb T,n}(L)$ denote the number of $n$-edge embeddings of $L$ in $\mathbb T$ having at least one vertex in the $x=0$ plane, and let $f_L$ be the number of prime factors of $L$. Tube-polymer link-statistics conjecture. There exist constants independent of $n$, but potentially dependent on $M_1$, $M_2$, and $L$, such that

$$p_{\mathbb T,n}(L)=A_{\mathbb T,L}n^{\alpha_{\mathbb T,L}}(\mu_{\mathbb T,0_1})^n(1+o(1)),\qquad n\to\infty,$$

where $\alpha_{\mathbb T,0_1}=0$, $A_{\mathbb T,L}>0$, $\mu_{\mathbb T,0_1}>0$, and

$$\alpha_{\mathbb T,L}=\alpha_{\mathbb T,0_1}+f_L.$$

This conjecture extends the knot-statistics asymptotics from polygons to arbitrary non-split links in tubes. Numerical evidence supports it for several tube sizes, while the general statement remains open.