Simplicial polytopes in dimension 4

Prove that the universality result for neighborly (hence simplicial in even dimensions) polytopes extends to dimension 4, implying ER-completeness (or an analogous universality) in dimension 4.

Background

Adiprasito and Padrol established universality for neighborly polytopes, yielding ER-completeness for simplicial polytopes in even dimensions. Dimension 4 is a pivotal boundary case for strengthening these results.

Encouraged by earlier universality in 4D (Richter-Gebert; Bokowski and Guedes de Oliveira), they explicitly conjecture an extension.

References

Encouraged by the results of Richter-Gebert and Bokowski and Guedes de Oliveira, Adiprasito and Padrol make the ``daring conjecture'' that the result holds in dimension 4.

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