Explanation of the box realization of the superline-graph construction

Determine why the superline-graph construction used in the paper should be expected to admit a realization as an intersection graph of boxes in $\mathbb{R}^3$.

Background

The paper constructs a triangle-free graph as a superline-graph and then gives an explicit realization of that graph as an intersection graph of boxes in three-dimensional space. The authors explain that the combinatorial origin of the graph is understood through the superline-graph framework, but they do not identify a conceptual reason that this particular construction should have such a geometric realization. This is presented as an unresolved explanatory aspect of the construction rather than as a missing theorem needed for the main results.

References

However, the part that is unclear to us is why one should expect $G$ to be realizable as an intersection graph of boxes in $\mathbb{R}3$.

— Geometric triangle-free graphs of large chromatic number  (2610.03517 - Tomon, 2 Oct 2026) in Discussion following the proof of Theorem 2.1, Section 2 (“Box-intersection graphs in dimension 3”)