Rota's unimodality conjecture for Whitney numbers of the first kind

Let $L$ be a geometric lattice of rank $r$. The Whitney numbers of the first kind are the integers $w_i(L)$ for $0\leq i\leq r$. Rota's unimodality conjecture. For some $0\leq k\leq r$,

$$w_0(L)\leq\dots\leq w_{k-1}(L)\leq w_k(L)\geq w_{k+1}(L)\geq\dots\geq w_r(L).$$

This is one of the famous unimodality conjectures for geometric lattices. The paper presents it as an unresolved direction connected with its topological interpretation of Whitney numbers; no resolution is supplied in the given text.