Zero-boundary-measure conjecture for SRS tiles

Let $\mathbf{r}=(r_0,\ldots,r_{d-1})\in\mathcal{E}_d$ with $r_0\neq0$, and let $\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ be the SRS tile associated with $\mathbf{x}\in\mathbb{Z}^d$. Write $\partial\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ for its boundary and let the $d$-dimensional Lebesgue measure be denoted by $\operatorname{Leb}_d$.

SRS tile boundary conjecture. For every $\mathbf{x}\in\mathbb{Z}^d$,

$$\operatorname{Leb}_d\bigl(\partial\mathcal{T}_{\mathbf{r}}(\mathbf{x})\bigr)=0.$$

This conjecture concerns the regularity of SRS tiles and is stated immediately after the weak-tiling conjecture. The survey gives no resolution.