Let $F_n$ be the free group of finite rank $n$ and let $\mathcal{O} \in \text{Out}(F_n)$. Let $\Gamma$ be a finite graph and $f : \Gamma \to \Gamma$ a relative train track representative of $\mathcal{O}$ with a maximal filtration $\emptyset = \Gamma_0 < \Gamma_1 < \cdots < \Gamma_k = \Gamma$. We have that $f(\Gamma_1) \subseteq \Gamma_1$ and the corresponding transition matrix is irreducible.
Is it always possible to find a relative train track representative of $\mathcal{O}$ such that $\mathcal{O}|_{\pi_1(\Gamma_1)} \in \text{Out}(\pi_1(\Gamma_1))$ is an irreducible outer automorphism?
