Generalized tame-ball conjecture for three-dimensional self-affine tiles

Let $p,q,r$ be integers with absolute value at least $2$, let $s,t,a_i,b_j$ be real numbers, and define

$$A=\begin{pmatrix}p&0&0\\0&q&0\\-t&-s&r\end{pmatrix}$$

and

$$\mathcal{D}=\{(i,j,k+a_i+b_j):0\le i<|p|,\ 0\le j<|q|,\ 0\le k<|r|\}.$$

Here $a_\infty$ and $b_\infty$ denote the limiting quantities used in the source's digit parametrization. Generalized tame-ball conjecture. If, for all $i,j$,

$$\left|\frac{a_{|p|-1}-a_\infty+t}{r(r-1)}+\frac{a_i-a_{i+1}}{r}\right|+\left|\frac{b_{|q|-1}-b_\infty+s}{r(r-1)}+\frac{b_j-b_{j+1}}{r}\right|<1,$$

then $T=T(A,\mathcal{D})$ is a tame ball in $\mathbb{R}^3$. This proposed extension is motivated by the paper's iterative reconstruction and homeomorphism techniques; its resolution is not established in the supplied text.