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

Let $L$ be a geometric lattice of rank $r$. The Whitney numbers of the second 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 the companion unimodality conjecture for geometric lattices, raised alongside the first-kind version in the paper's discussion of future directions. The supplied text gives no evidence of a resolution.