Extremality of complete-graph graphic matroids

Prove that, for every n>=1, the graphic matroid of the complete graph on n vertices is extremal.

Background

The paper seeks meaningful classes of extremal matroids, where extremality means that the associated point in the polytope Omega_{r,n} is a vertex. Complete-graph graphic matroids, also called braid matroids, are suggested as a natural class because existing valuative-invariant characterizations appear relevant. However, the cited characterization does not directly arise from a sequence of valuative invariants whose maxima certify a vertex, so the conjecture remains unresolved.

References

For every $n\geq 1$, the graphic matroid $\mathsf{K}_n$ of the complete graph on $n$ vertices is an extremal matroid.

— The polytope of all matroids  (2502.20157 - Ferroni et al., 27 Feb 2025) in Conjecture, Section 7.1, subsection “Extremal matroids and the pursuit of counterexamples in matroid theory”