Periodic tiling conjecture for translational monotile tilings

Let $T$ be a tile in $\mathbb{Z}^n$. A tiling by translated copies of $T$ is periodic if it is invariant under a nonzero translation of $\mathbb{Z}^n$. Periodic tiling conjecture. If $T$ tiles $\mathbb{Z}^n$ with translated copies, then it can tile $\mathbb{Z}^n$ periodically with translated copies. This conjecture would imply decidability of translational tiling by a single tile in each fixed dimension. It was disproved by Greenfeld and Tao, who constructed a tile admitting a translational tiling but no periodic translational tiling.