The TN-poset to UMEL-shellability conjecture for geometric lattices

Let $\mathrm{M}$ be a matroid. A TN-poset is a lower rank-uniform poset whose lower triangular matrix of lower Whitney numbers of the second kind is totally nonnegative, and UMEL-shellable refers to the shellability property defined in the source. The TN-poset to UMEL-shellability conjecture. If $\mathcal{L}(\mathrm{M})$ is a TN-poset, then $\mathcal{L}(\mathrm{M})$ is UMEL-shellable. The converse is false, as partition lattices are UMEL-shellable but not TN-posets; the forward implication is supported by rank-uniformity results, but the source gives no proof or resolution.