The SOP connective-constant conjecture

Let $w_{2n}$ be the number of self-overlapping polygons of length $2n$ in the SOP model, counted with multiplicity factor $\mu_\gamma$. SOP connective-constant conjecture. The limit

$$\lim_{n\to\infty}(w_{2n})^{1/(2n)}=1/g_*$$

exists and is finite. The analogous limit is proved for the uniform-measure model $\mathrm{SOP}_{\mathrm u}$, whereas the multiplicity-weighted SOP case is left conjectural.