The SRS weak-tiling conjecture

Let $\mathbf{r}=(r_0,\ldots,r_{d-1})\in\operatorname{int}(\mathcal{D}_d)$ with $r_0\ne0$. Weak-tiling conjecture. The collection

$$\{\mathcal{T}_{\mathbf{r}}(\mathbf{x})\mid \mathbf{x}\in\mathbb{Z}^d\}$$

is a weak tiling, meaning a weak $1$-tiling: every inner point belongs to exactly one tile. This is the stronger form of the preceding multiple-tiling conjecture, obtained by conjecturing that $m$ can always be chosen equal to one; the source gives no resolution in the supplied passage.