The periodic square peg problem

Let $L>0$, and let $\mathtt{Cyl}_L$ be the cylinder obtained from $\mathbb{R}^2$ by identifying points differing by $(L,0)$. Let $\mathtt{Graph}_{0,L}\colon \mathbb{R}/L\mathbb{Z}\to\mathtt{Cyl}_L$ be the standard closed curve $t\mapsto(t,0)$. Let $\sigma_1,\sigma_2\colon\mathbb{R}/L\mathbb{Z}\to\mathtt{Cyl}_L$ be simple curves homologous to $\mathtt{Graph}_{0,L}$, with disjoint images.

Periodic square peg conjecture. The union $\sigma_1(\mathbb{R}/L\mathbb{Z})\cup\sigma_2(\mathbb{R}/L\mathbb{Z})$ inscribes a square.

This is a periodic variant of Toeplitz's problem designed to avoid the issue of arbitrarily small squares. It is presented as an open problem in the paper.