Five-link characterization of the Reinhardt minimizer

Let $K$ be a hexameral domain with multi-curve $\sigma$ around its boundary. Let $t_0,t_1$ be the parameters defining the non-repeating boundary interval. Suppose that the entire non-repeating boundary $[t_0,t_1]$ is a hyperbolic chain with five links. Five-link conjecture. The equality $\mathbb\delta(K)=\delta_{\min}$ holds exactly when $K$ is the smoothed octagon, up to a transformation by $SL_2(\mathbb R)$. This is presented as a special case of the Reinhardt conjecture. Its resolution is not given in the supplied source.