Moebius rigidity of invariant metrics in boundaries of symmetric spaces of rank 1

Let $\mathrm{\partial }{\mathbf{H}}_{\mathbb{K}}^n$ 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 $\mathrm{\partial }{\mathbf{H}}_{\mathbb{K}}^n$. We prove that if $d$ is a metric on $\mathrm{\partial }{\mathbf{H}}_{\mathbb{K}}^n$ such that all Heisenberg similarities are $d$-M"obius maps, then under a topological condition $d$ is a constant multiple of a power of $d_{\mathfrak{H}}$.