Suppose that $\gamma$ is a curve in $\mathbb{C}\,$, with endpoints at zeros of $p$, and otherwise missing the zeros of $p$, such that its pointwise phase $\theta$ (6.4) satisfies (7.1) for all Lagrangians $L_{i}=\gamma^{n}_{i},\ i=1,2,$ fibred over curves $\gamma_{i}$ in the base, and also $S-I:=\sup_{\gamma}\theta-\inf_{\gamma}\theta<2\pi/3$. Then the flow (6.9) exists for all time and converges in $C^{\infty}$ to a smooth curve whose phase function (6.4) is constant.