Fixed points of irreducible, displacement one automorphisms of free products

We consider the action of outer automorphisms on the deformation space $\mathcal{O}$ of $G$-trees given by a free product decomposition of a group $G$. We show that an irreducible, displacement 1 automorphism fixes exactly one point in ${\mathcal{O}}_1$ (the covolume 1 slice of $\mathcal{O}$).