The sparse paving characterization by Schubert coefficients

Let $M$ be a connected matroid of rank $r$ on $[n]$. Write $U_{r,n}$ for the uniform matroid of rank $r$ on $[n]$, let $d_\lambda(M)$ denote the Schubert coefficient indexed by a partition $\lambda$, and let $h^c$ denote the complement of the hook partition $h$. Sparse paving characterization. $M$ is sparse paving if and only if

$$d_\lambda(M)=d_\lambda(U_{r,n})$$

for all partitions $\lambda\neq h^c$. The paper proves the forward implication for connected sparse paving matroids; the converse remains conjectural.