Adjacency of vertices in the matroid polytopes

Determine when two matroids yield neighboring vertices of the labelled polytope \overline{\Omega}_{r,n}, and when two isomorphism classes of extremal matroids yield neighboring vertices of the unlabelled polytope \Omega_{r,n}.

Background

After analyzing vertices and several faces of the labelled and unlabelled polytopes, the paper identifies the edge structure as an unresolved direction. For the labelled polytope, the question concerns adjacency among all matroid vertices. For the unlabelled polytope, it concerns only isomorphism classes that are extremal, and is described as substantially more difficult because the extremal matroids are not yet well understood.

References

The following question about the edges intrigues us. When do two matroids $M$ and $N$ give a pair of neighboring vertices of $\overline{\Omega}{r,n}$? When do two isomorphism classes of extremal matroids yield neighboring vertices of $\Omega{r,n}$?

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Question, Section 7.3, subsection “Edges of the polytopes”