Inclusion of unit d-interval graphs in disjoint balanced d-interval graphs for d > 2

Determine whether, for every integer d > 2, the class of unit d-interval graphs is contained within the class of disjoint balanced d-interval graphs.

Background

The paper establishes that for d = 2, unit 2-interval graphs are properly contained in disjoint balanced 2-interval graphs, but shows that certain relationships do not generalize straightforwardly for larger d.

The authors explicitly point out that it remains unresolved whether the inclusion unit d-interval ⊆ disjoint balanced d-interval continues to hold for d > 2.