Lyapunov-exponent upper bound for permanent saturation of metric graphs

Let $G=(V,E,l)$ be a connected metric graph endowed with an interval exchange transformation $T$ partitioning $G$ into $n$ intervals with lengths $\ell_1, \ell_2, \ldots, \ell_n$ linearly independent over $\mathbb{Q}$. Suppose $\mu$ is the Lebesgue measure preserved by $T$. Let $\tau_s$ denote the saturation time, let $\varepsilon$ be the accuracy parameter, and let $K$ be a graph-dependent constant. Let $\lambda_1 \geq \dots \geq \lambda_{n-1}$ be the non-negative Lyapunov exponents of $T$. Lyapunov-exponent saturation conjecture. The saturation time satisfies

$$\tau_s \leq \frac{K \cdot \exp\left(\sum_{i=1}^{n-1}\lambda_i\right)}{\varepsilon}\left(\log\left(K \cdot \exp\left(\sum_{i=1}^{n-1}\lambda_i\right)\right)-\log\varepsilon\right)^{n-1}.$$

The conjectural bound is motivated by the proposed relationship between saturation time and Lyapunov-exponent growth, but the source reports only numerical support and does not establish the inequality.