The price of anarchy in the max-distance network creation game is not constant

At edge price $\alpha =1$, we construct an infinite family of pure Nash equilibria of the unilateral max-distance network creation game with $\text{PoA}\ge 2^{\sqrt{{\log }_{2}n}-O(\log \log n)}$. Together with the known upper bound, this gives $2^{\mathrm{\Theta }(\sqrt{\log n})}$ along the constructed sequence of population sizes. We subdivide every edge of the bipartite double cover of a distance-uniform graph with large diameter constructed by Lavrov, Loh and Messegué, and let each subdivision vertex buy its two incident edges. A distance calculation rules out every profitable unilateral deviation. The equilibria are not strict. We also give a short proof that the price of anarchy is constant for every polynomially vanishing edge price.