The maximally efficient surface braid conjecture

Let $\mathbf{G}$ be the countably infinite set of regular maps on the torus, where each $\mathcal{G}\in\mathbf{G}$ is an embedded graph in the torus that is arc-transitive. For each $\mathcal{G}\in\mathbf{G}$, let $\mathbf{B}_{\mathcal{G}}$ be the set of braid words formed from any number of braid generators defined on $\mathcal{G}$, and let $\overline{h}(\beta)$ denote the topological entropy per operation of $\beta\in\mathbf{B}_{\mathcal{G}}$. Maximally efficient surface braid conjecture.

$$\max_{\beta\in\mathbf{B}_{\mathcal{G}},\,\mathcal{G}\in\mathbf{G}}\overline{h}(\beta)=\log(\phi+\sqrt{\phi}),$$

where

$$\phi=\frac{1+\sqrt{5}}{2}.$$

Furthermore, when extended to the plane using the periodicity of the pertinent torus model, all braid words $\beta$ for which $\overline{h}(\beta)=\log(\phi+\sqrt{\phi})$ represent the same braid up to conjugacy and time-reversal symmetry. This conjecture asserts both the maximal topological entropy per operation among braid words on regular torus maps and uniqueness, up to the stated symmetries, of the maximizers.