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

At edge price $α=1$, we construct an infinite family of pure Nash equilibria of the unilateral max-distance network creation game with $\PoA\ge2^{\sqrt{\log_2 n}-O(\log\log n)}$. Together with the known upper bound, this gives $2^{Θ(\sqrt{\log n})}$ along the constructed sequence of population sizes