The complete-graph graphic-matroid extremality conjecture

For each integer $n\geq 1$, let $\mathsf{K}_n$ be the graphic matroid of the complete graph on $n$ vertices. A matroid is extremal if it is an extremal matroid in the polytope studied in the source.

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

The conjecture is motivated by a theorem of Bonin and Miller characterizing these graphic matroids through valuative invariants, although that characterization does not directly establish extremality. The source suggests that modifying their strategy may lead to a proof.