Speyer's conjecture on matroid subdivision face numbers

Let $\mathcal{S}$ be a subdivision of $\Delta_{k,n}$ into smaller matroid polytopes. For each $1\leq i\leq n$, let $f_i$ be the number of cells of $\mathcal{S}$ of dimension $n-i$ lying in the interior of $\Delta_{k,n}$. Speyer's conjecture. One has

$$f_i \leq \binom{n-1-i}{k-i}\binom{n-k-1}{i-1}.$$

Moreover, simultaneous equality holds if and only if all facets of $\mathcal{S}$ correspond to base polytopes of series-parallel matroids. This conjecture predicts sharp bounds on the face numbers of matroid subdivisions of the hypersimplex; its resolution status is not specified in the source.