Periodic orbit-closure conjecture for exact-cluster tilings

Let $F\subseteq\mathbb{Z}^2$ be an exact cluster with full affine span, and let $T$ be an $F$-tiling. The orbit closure of $T$ is

$$\overline{\mathbb{Z}^2\cdot T}=\operatorname{cl}\{v\cdot T:v\in\mathbb{Z}^2\}.$$

A tiling is $1$-periodic if it has a nonzero period in $\mathbb{Z}^2$.

Periodic orbit-closure conjecture. The orbit closure

$$\overline{\mathbb{Z}^2\cdot T}$$

contains a $1$-periodic $F$-tiling.

This conjecture would extend the expected link between low pattern complexity and periodicity to tilings by possibly non-convex windows. The paper states that its counterexample refutes this conjecture, so it is disproved.