The semi-cross lattice tiling conjecture

Let $p$ be a prime, and define the semi-cross

$$V_{p-1}=\{\mathbf{0},\mathbf{e}_1,\ldots,\mathbf{e}_{p-1}\}\subset\mathbb{Z}^{p-1}.$$

Semi-cross lattice tiling conjecture. Every tiling of $\mathbb{Z}^{p-1}$ by $V_{p-1}$ is lattice. The paper states that this special case is equivalent to the prime-tile lattice tiling conjecture, via the universal semi-cross construction.