The incline angle of the tangent vector of $F(\partial\Sigma)$ is equal to the Lagrangian angle mod $\pi\mathbb{Z}$. Together with Claim 3.22 this implies $|\arg F|\leq\frac{\pi}{2}-\epsilon$ on the $\partial\Sigma$, whence the same bound holds on $\Sigma$ by the maximum principle for holomorphic functions. ∎