Existence of maximal cells in Grassmann polytopes

Let $P$ be a nonempty Grassmann polytope. A maximal cell is an independent set $f$ in the relevant triangulation data such that $f\leq h$ and

$$\dim(Z(\Pi_{f,\geq 0}))=\dim(P).$$

Maximal-cell conjecture. Every nonempty Grassmann polytope has a maximal cell.

This is a basic existence assertion needed for the proposed triangulation theory; no proof or resolution is supplied in the given text.