Stabilization conjecture for chromatic numbers of rank-r Abelian Cayley graphs

Let $r$ be a positive integer. Let $M_X$ be an $m\times r$ integer matrix and let $X=M_X^{SACG}$ denote the Abelian Cayley graph associated with $M_X$. Assume that $M_X$ has no zero rows, and that $X$ is not bipartite and has no loops. Stabilization conjecture. For all sufficiently large $m$, one has

$$\chi(X)=3.$$

The paper proves the analogous statement for rank two, where the chromatic number stabilizes starting at dimension $5$. The conjecture proposes that the same stabilization occurs for every fixed positive rank, while the corresponding higher-rank cases remain open.