Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are there known estimates on the minimum distortion of a bi-Lipschitz embedding of $B_{\mathbb{H}^n}(x,r)$ into $\mathcal{W}_2(\mathbb{R})$.

Let me just note that a bi-Lipschitz embedding into $\mathcal{W}_2(\mathbb{R})$ must exist by this paper.