The area inequality for four periodic curves

Let $L>0$, and let $\mathtt{Cyl}_L$ and $\mathtt{Graph}_{0,L}$ be as above. Let $\sigma_1,\sigma_2,\sigma_3,\sigma_4\colon\mathbb{R}/L\mathbb{Z}\to\mathtt{Cyl}_L$ be simple closed polygonal paths homologous to $\mathtt{Graph}_{0,L}$. Assume that the ordered quadruplet of their images does not jointly inscribe a square.

Area inequality conjecture. Then

$$\int_{\sigma_1}y\,dx-\int_{\sigma_2}y\,dx+\int_{\sigma_3}y\,dx-\int_{\sigma_4}y\,dx\ne0.$$

The claim is a stronger statement related to the quadripartite periodic square peg problem. The paper presents it as conjectural for polygonal curves.