Continuity of the connective constant under local convergence

Let $G$ be an infinite transitive connected graph, and let $\{G_n\}_{n=1}^{\infty}$ be a sequence of infinite transitive connected graphs that converges locally to $G$. For each graph, let $\mu(G)$ denote its connective constant, defined as the exponential growth rate of the number of self-avoiding walks.

Continuity conjecture. The connective constant is continuous with respect to local convergence:

$$\lim_{n\to\infty}\mu(G_n)=\mu(G).$$

The connective constant governs the exponential growth of self-avoiding walks and is known explicitly for very few graphs. Continuity under local convergence would allow connective constants of locally approximating transitive graphs to determine that of the limit; the source provides no resolution of this assertion.