The SRS boundary measure conjecture

Let $\mathbf{r}=(r_0,\ldots,r_{d-1})\in\operatorname{int}(\mathcal{D}_d)$ be a reduced parameter, and let $\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ be the associated SRS tile for $\mathbf{x}\in\mathbb{Z}^d$. SRS boundary measure conjecture. For every $\mathbf{x}\in\mathbb{Z}^d$, the boundary $\partial\mathcal{T}_{\mathbf{r}}(\mathbf{x})$ has zero $d$-dimensional Lebesgue measure. The source states that this is known for parameters associated with Pisot units and monic CNS, while it remains open in general.