Lee–Naor nonembedding conjecture for the Heisenberg group

Let $(\mathbb H,d^{\mathbb H})$ be the Heisenberg group equipped with its Carnot–Carathéodory metric. Lee–Naor conjecture. The metric space $(\mathbb H,d^{\mathbb H})$ does not admit a bi-Lipschitz embedding into $L^1$. This conjecture concerns the failure of bi-Lipschitz embeddability into $L^1$ for the Heisenberg group; the source states that it is proved in the present paper.