Deng–Liu–Ngai conjecture on ball-like self-affine tiles

Let $d\ge 2$, let $p_1,\ldots,p_d$ be integers with $|p_i|\ge 2$ for $1\le i\le d$, let $A_1=\operatorname{diag}(p_1,\ldots,p_{d-1})$, and let $\boldsymbol{s}=(s_1,\ldots,s_{d-1})\in\mathbb{R}^{d-1}$. Suppose the self-affine pair $(A,\mathcal{D})$ satisfies

$$A=\begin{pmatrix}A_1&-\boldsymbol{s}\\0&p_d\end{pmatrix},\qquad \mathcal{D}=\{(i_1,\ldots,i_d):0\le i_j<|p_j|,\ 1\le j\le d\}.$$

Deng–Liu–Ngai conjecture. Under these hypotheses, the self-affine tile $T$ is homeomorphic to a $d$-dimensional ball if and only if its interior $T^\circ$ is connected. The conjecture concerns whether contractibility, which follows when $T^\circ$ is connected, suffices for the tile to be a topological ball; the paper's results answer it positively for this class.