Recognition of treetope graphs in higher dimensions

Determine the computational complexity of recognizing treetope graphs in dimensions five and higher.

Background

Treetopes are special polytopes with a distinguished base facet intersecting all higher-dimensional faces. Recognition in 4D is polynomial-time solvable, but no classification is known for ≥ 5D.

Understanding this would bridge polyhedral graph recognition across dimensions and connect to ER-completeness conjectures.

References

The complexity of this problem is open for dimension five and above.

— The Existential Theory of the Reals as a Complexity Class: A Compendium (2407.18006 - Schaefer et al., 25 Jul 2024) in Compendium — Problem 'Treetope graph (Open)'