The sign-stability estimate for pA mutation loops

Let $\phi$ be a pA mutation loop on a marked surface $\Sigma$. Write $\mathcal{E}_\phi^a$ for its associated asymptotic entropy and $\lambda_{\pi(\phi)}$ for the spectral radius of the mapping class induced by $\phi$. Sign-stability estimate. For any pA mutation loop $\phi$ on a marked surface $\Sigma$, we have

$$\mathcal{E}_\phi^a \geq \log\lambda_{\pi(\phi)}.$$

This estimate is presented as a natural expectation in the discussion of weakly sign-stable mutation loops; the supplied text gives no proof or resolution.