  Let $\partial{\bf H}^n_{\mathbb K}$ denote the boundary of a symmetric space
of rank-one and of non-compact type and let $d_{\mathfrak{H}}$ be the Kor\'anyi
metric defined in $\partial{\bf H}^n_{\mathbb K}$. We prove that if $d$ is a
metric on $\partial{\bf H}^n_{\mathbb K}$ such that all Heisenberg similarities
are $d$-M\"obius maps, then under a topological condition $d$ is a constant
multiple of a power of $d_{\mathfrak{H}}$.
