The surface-independence conjecture for asymptotic algebraic connectivity

Fix a surface $S$, and let $\mathcal{A}^{\infty}(S)$ denote the asymptotic algebraic connectivity associated with graphs on $S$. The double wheel graph is a graph whose algebraic connectivity is two. Surface-independence conjecture. The value of $\mathcal{A}^{\infty}(S)$ is independent of the fixed surface $S$, and

$$\mathcal{A}^{\infty}(S)\equiv 2.$$

This conjecture predicts that the general asymptotic upper bound is not optimal and that the common value is the algebraic connectivity of the double wheel graph.