On the geometry of the free factor graph for $\mathrm{A}\mathrm{u}\mathrm{t}(F_N)$

Let $\mathrm{\Phi }$ be a pseudo-Anosov diffeomorphism of a compact (possibly non-orientable) surface $\mathrm{\Sigma }$ with one boundary component. We show that if $b\in {\pi }_1(\mathrm{\Sigma })$ is the boundary word, $\phi \in \mathrm{A}\mathrm{u}\mathrm{t}({\pi }_1(\mathrm{\Sigma }))$ is a representative of $\mathrm{\Phi }$ fixing $b$, and ${\mathrm{a}\mathrm{d}}_b$ denotes conjugation by $b$, then the orbits of $⟨\phi ,{\mathrm{a}\mathrm{d}}_b⟩\cong {\mathbb{Z}}^2$ in the graph of free factors of ${\pi }_1(\mathrm{\Sigma })$ are quasi-isometrically embedded. It follows that for $N\ge 2$ the free factor graph for $\mathrm{A}\mathrm{u}\mathrm{t}(F_N)$ is not hyperbolic, in contrast to the $\mathrm{O}\mathrm{u}\mathrm{t}(F_N)$ case.