Clockwise along $\partial\Sigma$, the function $\text{Re }F$ is decreasing on the $L$ boundary portion, but increasing on the $L^{\prime}=L_{1}\cup L_{2}$ boundary portion. In particular,

$$0=\text{Re }F(q)\geq\text{Re }F\geq\text{Re }F(p).$$

More intrinsically, the real part of the complex volume forms on the moduli spaces at such $u:\Sigma\to X$ are nonpositive.