Consistency conjecture for adjacent distance graphs

Let $n\geq 1$ be a number. Let $\Delta_n$ be the graph on $\mathbb{R}^n$ connecting points of rational Euclidean distance. The adjacent distance-graph consistency conjecture. The statement that the chromatic number of $\Delta_n$ is countable while that of $\Delta_{n+1}$ is not is consistent with $\mathrm{ZF}+\mathrm{DC}$. This concerns the possibility of separating the chromatic behavior of consecutive distance graphs in choiceless set theory; the conjecture is still open for $n=4$ and higher.