Characterization of extremal discrete Borsuk sets by cubes

Let $S \subset \mathbb{Z}^d$ be a bounded set, and let $\beta_{\mathbb{Z}}(S)$ denote its discrete Borsuk partition number. Let $\operatorname{conv}(S)$ be the convex hull of $S$, and call two lattice polytopes unimodularly equivalent if one is mapped to the other by an affine unimodular transformation. Discrete cube characterization conjecture.

$$\beta_{\mathbb{Z}}(S)=2^d$$

if and only if $\operatorname{conv}(S)$ is unimodularly equivalent to a $d$-cube $[0,m]^d$ for some $m \in \mathbb{N}$. The paper proves the upper bound $\beta_{\mathbb{Z}}(S)\leq 2^d$ and that it is attained by $S=\{0,1\}^d$; the conjecture seeks to characterize all equality cases.