The symmetric three-curve area inequality

Let $L>0$, and let $\mathtt{Cyl}_L$ be the cylinder obtained from $\mathbb{R}^2$ by identifying points differing by $(L,0)$, with standard curve $\mathtt{Graph}_{0,L}$. Let $\gamma_1,\gamma_2,\gamma_3\colon\mathbb{R}/L\mathbb{Z}\to\mathtt{Cyl}_L$ be simple closed polygonal paths homologous to $\mathtt{Graph}_{0,L}$. Assume there do not exist $x,y_1,y_2,y_3\in\mathbb{R}$ such that $(x,y_i)\in\gamma_i(\mathbb{R}/L\mathbb{Z})$ for all $i$ and $y_1+y_2+y_3=0$.

Symmetric area inequality conjecture. One has

$$\int_{\gamma_1+\gamma_2+\gamma_3}y\,dx\ne0.$$

This is the one-dimensional symmetric reformulation of the special case of the area inequality with one curve equal to $\mathtt{Graph}_{0,L}$. It is related to a combinatorial problem on non-crossing sums and remains open in the paper.