Let ∂HKn​ denote the boundary of a symmetric space
of rank-one and of non-compact type and let dH​ be the Kor\'anyi
metric defined in ∂HKn​. We prove that if d is a
metric on ∂HKn​ such that all Heisenberg similarities
are d-M\"obius maps, then under a topological condition d is a constant
multiple of a power of dH​.Comment: Third version, 13 pages. Contains simpler and shortened proof