Combinatorial-geometric facet equality conjecture for Grassmann polytopes

Let $P$ be a Grassmann polytope, let $F$ be its global combinatorial facet, and let $Z(F)_{\geq 0}$ be the union of the Grassmann polytopes indexed by $F$. Let $Z(F)$ be its Zariski closure, and let $P\cap Z(F)$ be the corresponding global geometric facet.

Facet equality conjecture.

$$Z(F)_{\geq 0}=P\cap Z(F),$$

so that combinatorial and geometric facets agree.

The claim would identify the combinatorial union of cells with the geometric facet cut out by its Zariski closure.