The quadripartite 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 $t\mapsto(t,0)$. For four subsets of the cylinder, say that they jointly inscribe a square if their Cartesian product intersects the closure of the space of squares in $\mathtt{Cyl}_L$; equivalently, there are $x\in\mathbb{R}/L\mathbb{Z}$ and $y,a,b\in\mathbb{R}$ such that the four points $(x,y)$, $(x+a,y+b)$, $(x+a-b,y+a+b)$, and $(x-b,y+a)$ lie in the four sets in order. Let $\sigma_1,\sigma_2\colon\mathbb{R}/L\mathbb{Z}\to\mathtt{Cyl}_L$ be simple closed curves homologous to $\mathtt{Graph}_{0,L}$.

Quadripartite periodic square peg conjecture. The quadruplet

$$\bigl(\sigma_1(\mathbb{R}/L\mathbb{Z}),\sigma_1(\mathbb{R}/L\mathbb{Z}),\sigma_2(\mathbb{R}/L\mathbb{Z}),\sigma_2(\mathbb{R}/L\mathbb{Z})\bigr)$$

jointly inscribes a square.

This claim would imply the periodic square peg problem and allows degenerate inscribed squares through the closure formulation. It is presented as an open conjecture.