  Let $G$ be a locally compact group and $\mu$ an admissible probability
measure on $G$. Let $(B,\nu)$ be the universal topological Poisson
$\mu$-boundary of $(G,\mu)$ and $\Pi_s(G)$ the universal minimal strongly
proximal $G$-flow. This note is inspired by a recent result of Hartman and
Kalantar. We show that for a locally compact second countable group $G$ the
following conditions are equivalent: (i) the measure $\nu$ is the unique
$\mu$-stationary measure on $B$, (ii) $(B,G) \cong (\Pi_s(G),G)$.
