  This paper studies the geometric rigidity of the universal Coxeter group of
rank $n$, which is the free product $W_n$ of $n$ copies of
$\mathbb{Z}/2\mathbb{Z}$. We prove that for $n\geq 4$ the group of symmetries
of the spine of the Guirardel-Levitt outer space of $W_n$ is reduced to the
outer automorphism group $\mathrm{Out}(W_n)$.
