Chari's shellable $M$-vector conjecture for matroid complexes

A matroid complex is the simplicial complex of independent sets of a matroid. For a $(d-1)$-dimensional simplicial complex $\Delta$, its $h$-vector $h(\Delta)=(h_0,h_1,\ldots,h_d)$ is defined by

$$\sum_{i=0}^{d}h_i y^{d-i}=\sum_{i=0}^{d}f_i(y-1)^{d-i},$$

where $f_i$ counts the faces of cardinality $i$. A monomial order ideal is a set of monomials closed under divisibility, and an $M$-vector is the degree sequence of such an ideal. A shellable $M$-vector is the degree sequence arising from an $M$-shellable poset, equivalently from a monomial structure admitting the shelling condition defined in the source. Chari's conjecture. The $h$-vector of a matroid complex is a shellable $M$-vector. Chari's statement is described as a stronger version of Stanley's conjecture, and the paper's abstract says that it gives a positive answer in a partial case; the source paper proves that every discrete polymatroid is $M$-shellable.