Speyer's tropical $f$-vector conjecture

Let $\Sigma(r,n)$ be the hypersimplex, and consider a subdivision of $\Sigma(r,n)$ into matroid base polytopes. An interior face is a face of the subdivision not contained in the boundary of $\Sigma(r,n)$. Speyer's tropical $f$-vector conjecture. The number of $(n-i)$-dimensional interior faces in such a subdivision is at most

$$\frac{(n-i-1)!}{(r-i)!\,(n-r-i)!\,(i-1)!}.$$

This conjecture concerns the face numbers of subdivisions of hypersimplices into matroid base polytopes, which arise in the study of tropical linear spaces. It is proved in the paper via the Cohen–Macaulayness of the external activity complex and the resulting formulas for its $K$-polynomial and related matroid invariants.