Quasi-Möbius characterization of snowflakes of Heisenberg groups

Let $\mathcal{A}=\{\mathsf{R},\mathsf{C},\mathsf{H},\mathsf{O}\}$ be the collection of normed division algebras. For ${\mathsf K}\in\mathcal{A}$ and $n\in\mathsf N$, let $\mathbb{H}_{\mathsf K}^n$ be the $n$-th ${\mathsf K}$-Heisenberg group, equipped with its visual distance $\rho$. Let $X$ be a metric space and let $\alpha\in(0,1]$. Quasi-Möbius characterization conjecture. The metric space $X$ is bi-Lipschitz equivalent to $(\mathbb{H}_{\mathsf K}^n,\rho^\alpha)$ if and only if $X$ is locally compact, connected, bi-Lipschitz homogeneous, and quasi-invertible. This conjecture seeks a characterization, up to bi-Lipschitz equivalence, of snowflakes of boundaries of non-compact rank-one symmetric spaces using metric homogeneity and quasi-invertibility; the paper presents it as an ongoing goal, and no resolution is supplied here.