Necessary and sufficient condition for bounded distance equivalence of parallelotope model sets

Let $\Gamma \subset \mathbb{R}^m \times \mathbb{R}^n$ be a lattice, let $p_2$ denote projection onto the second factor, and let $W \subset \mathbb{R}^n$ be the half-open parallelotope

$$W = \left\{ \sum_{j=1}^n t_j v_j \,: \, 0 \leq t_j <1 \right\},$$

where $v_1, \ldots, v_n$ are linearly independent vectors in $\mathbb{R}^n$. The bounded-distance equivalence conjecture. The model set $\Lambda(\Gamma, W)$ is bounded distance equivalent to a lattice if and only if there exist vectors $w_1, \ldots, w_n \in p_2(\Gamma)$ such that

$$v_1 = w_1, \quad v_k = w_k + \operatorname{span}(w_1, w_2, \ldots, w_{k-1}) \quad (2\leq k \leq n).$$

This conjecture proposes that the sufficient condition for bounded distance equivalence established for parallelotope windows is also necessary, characterizing exactly when such a model set is bounded distance equivalent to a lattice.