Conjecture on the connective constant of universal covers of punctured hexagonal lattices

Let ${\mathbb U}^\infty$ be the universal cover of the hexagonal lattice $\mathbb H$ with a singularity at the origin, let $\pi_\infty$ be its canonical projection to $\mathbb H$, and let $\Lambda_k$ be the ball of radius $k$ in the face graph centred at the origin. Define

$${\mathbb U}^\infty_k:={\mathbb U}^\infty\setminus \pi_\infty^{-1}(\Lambda_k).$$

For $k\in\mathbb N$ and $v\in{\mathbb U}^\infty_k$, let $\mathsf{SAW}_n({\mathbb U}^\infty_k,v)$ be the set of self-avoiding walks of length $n$ in ${\mathbb U}^\infty_k$ starting at $v$, and define

$$\mu({\mathbb U}^\infty_k):=\lim_{n\to\infty}\sup_{v\in{\mathbb U}^\infty_k}\left|\mathsf{SAW}_n({\mathbb U}^\infty_k,v)\right|^{1/n}.$$

Universal-cover connective-constant conjecture. There exists $k\in\mathbb N\cup\{0\}$ for which

$$\mu({\mathbb U}^\infty_k)=\mu(\mathbb H).$$

This qualitative conjecture concerns the connective constant of self-avoiding walks on universal covers of the hexagonal lattice with increasingly large punctures. It is introduced as a hypothesis for obtaining a polynomial upper bound on the normalized number of self-avoiding walks; the supplied text gives no evidence that it has been resolved.