Adjacency in the matroid polytopes

Characterize when two matroids yield neighboring vertices of $\overline{\Omega}_{r,n}$ and when two isomorphism classes of extremal matroids yield neighboring vertices of $\Omega_{r,n}$; in particular, determine the edge structure of these polytopes.

Background

The paper develops the vertices and several faces of the labelled and unlabelled polytopes but does not characterize their edges. The unlabelled question is especially difficult because only extremal matroids correspond to vertices, and the paper has limited understanding of the full extremal class.

References

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 7.2, Section 7.3, “Edges of the polytopes”