Self-affine tile digit-set stabilization conjecture

Let $R$ be an $n\times n$ expanding integer matrix and let $0\in\mathcal D\subset\mathbb Z^n$ satisfy $\#\mathcal D=|\det(R)|$. Define $T(R,\mathcal D)$ to be the associated self-affine tile, let $\mathcal D_m$ denote the $m$-fold digit expansion, and set $\mathcal J_0^*=\mathbb Z^n$ and $\mathcal J_k^*=R\mathcal J_{k-1}^*+\mathcal D$ for $k\geq1$. Self-affine tile stabilization conjecture. If $T(R,\mathcal D)$ is a self-affine tile, then there exists an integer $m\geq1$ such that

$$\mathcal J_{m+1}^*=\mathcal J_m^*.$$

The paper states that this conjecture is true in the one-dimensional case, while the general higher-dimensional question is the subject of the conjecture.