Suppose you are given a directed graph $G=(V,E)$ which is strongly connected, i.e. for every two vertices $u,v \in V$ there exists a directed path between them. Consider the corresponding edge shift space $(X_G,\Sigma)$ with a Markov probability measure $\nu$ on $X_G$ and a measurable matrix cocycle $A: X_G\to \text{GL}(d,\mathbb{R})$ for some $d\ge 2$ which asigns an unimodular and non-negative integer matrix $M_{\gamma_0}$ to an infinite path $\gamma \in X_G$, i.e. $\det(M_{\gamma_0}) = \pm 1$ with $M_{\gamma_0} \ge 0$. Suppose aditionally that there exists a finite directed path $\gamma = \gamma_0 \gamma_1 \ldots \gamma_{l-1}$ in $G$ such that $\nu(_0[\gamma]) > 0$ and $M_{\gamma} = M_{\gamma_0}\cdot M_{\gamma_1} \cdot \ldots \cdot M_{\gamma_{l-1}} > 0$, then by a classical result due to H. Furstenberg we have that for $\nu$-almost every $\gamma \in X_G$, there exists a unique probability vector $\mu(\gamma)$ such that $\bigcap_{n\ge 1} M_{\gamma_{[0,n)}} \mathbb{R}^d_+ = \mathbb{R_+}\mu(\gamma)$, where $M_{\gamma_{[0,n)}} = M_{\gamma_0} \cdot \ldots \cdot M_{\gamma_{n-1}}$, if this is the case we say that $\mu(\gamma)$ is the generalized right Perron-Frobenius eigenvector associated with $\gamma$.
In Berthé et. al. the authors prove that if the Pisot conditions $\theta_1^{\nu} > 0 > \theta_2^{\nu}$ holds then for almost every $\gamma\in X_G$ the generalized right Perron-Frobenius eigenvector has rationally independent coordinates.
Question: Does there exist a distinct condition that ensures that for $\nu$-almost every $\gamma \in X_G$ is such that $\mu(\gamma)$ has rationally independent coordinates?
