Speyer's interior-face conjecture for matroid subdivisions

Let $M$ be a matroid on $[n]$ of rank $d$, and let a matroid subdivision of $M$ be a polyhedral subdivision of its matroid polytope in which every cell is itself a matroid polytope. Speyer's conjecture. For each $1\leq c\leq\min\{d,n-d\}$, such a subdivision has at most

$$\frac{(n-c-1)!}{(d-c)!(n-d-c)!(c-1)!}$$

interior faces of dimension $n-c$. Matroid subdivisions connect polyhedral combinatorics with tropical geometry and valuated matroids. The source gives no resolution evidence for this conjecture.