Fuglede conjecture for two non-overlapping unit cubes in higher dimensions

Determine whether Fuglede's conjecture holds for a union of two non-overlapping unit cubes in d, in higher dimensions and without assuming that the cubes are parallel to the coordinate axes.

Background

The paper proves that Fuglede's conjecture holds for a union of two non-overlapping unit squares in $\mathbb{R}^2$, including squares that are not necessarily parallel. The argument uses the non-spectrality theorem for unions containing non-axis-parallel polytopes together with the classification of weak tiles formed by two translates of the unit cube.

The authors explain that the higher-dimensional analogue remains unresolved because their non-spectrality theorem does not apply when an additional cube has even one facet parallel to a coordinate hyperplane. Thus, the unresolved problem concerns the full higher-dimensional configuration of two non-overlapping unit cubes, including cases with relative rotations that produce such parallel facets.