Existence of negatively self-polar polytopes with equidistant vertices

Determine for which values of the distance r and dimension d there exists a negatively self-polar polytope in R^d whose vertices are all at distance r from the origin.

Background

The paper relates strongly self-dual polytopes to negatively self-polar polytopes whose vertices lie on a common sphere centered at the origin. It notes that Jensen formulated an existence problem concerning negatively self-polar polytopes with this equidistance property. The problem asks for a complete characterization of the admissible pairs of vertex distance r and dimension d, and is not resolved in the paper.

References

Jensen formulated some questions, from which Question 9.5. an existence problem: For which values of $r$ and $d$ does there exist a negatively self-polar polytope in $Rd$ such that its vertices are $r$ away from the origin?

— Strongly self-dual polytopes  (2501.16121 - Horváth et al., 27 Jan 2025) in Section 1, Introduction