  This paper studies the rigidity properties of the abstract commensurator of
the outer automorphism group of a 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 $n\geq 5$ the natural map $\mathrm{Out}(W_n) \to
\mathrm{Comm}(\mathrm{Out}(W_n))$ is an isomorphism and that every isomorphism
between finite index subgroups of $\mathrm{Out}(W_n)$ is given by a conjugation
by an element of $\mathrm{Out}(W_n)$.
