The f-vector conjecture for matroidal decompositions of uniform matroids

Let $\mathrm{U}_{r,n}$ be the uniform matroid of rank $r$ on $[n]$, and let $\Delta(\mathrm{U}_{r,n})$ be its matroid base polytope. A matroidal decomposition is a decomposition of this polytope into matroid base polytopes. The f-vector conjecture. In any matroidal decomposition of $\Delta(\mathrm{U}_{r,n})$, the number of faces of dimension $n-i$ is at most

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

Equality occurs if and only if each polytope in the decomposition comes from a direct sum of series-parallel matroids. The conjecture is based on computations with tropical linear spaces and remains open.