Sharp $cl_1$ distortion conjecture for Heisenberg balls

Let $\mathbb H$ be the discrete Heisenberg group, let $B_n$ be its radius-$n$ ball with the word metric $d_W$, and let $c_1(B_n,d_W)$ denote the least distortion of an embedding of this metric space into $\ell_1$. Sharp $\ell_1$ distortion conjecture for Heisenberg balls. For every integer $n\geqslant2$,

$$c_1(B_n,d_W)\asymp\sqrt{\log n}.$$

The conjectural functional and discrete inequalities preceding this statement would imply the lower bound $c_1(B_n,d_W)\gtrsim\sqrt{\log n}$; the matching upper bound is already provided by known embeddings. Thus the conjecture predicts the asymptotically sharp distortion.