The Plücker-coordinate volume relation for polytopes

Let $G(n-d,n)$ denote the Grassmannian of $(n-d)$-dimensional subspaces of an $n$-dimensional space, and let $\bm{p}$ be a Plücker-coordinate vector in $\bigcup\limits_n G(n-d,n)$. Let $V$ be the volume of the corresponding polytope. Plücker-coordinate volume conjecture. There exists a map

$$f:\bigcup\limits_n G(n-d,n)\rightarrow\mathbb{R},\qquad \bm{p}\mapsto x$$

such that

$$V\sim |x|+1,$$

where $1$ indicates the smallest normalized volume among the polytopes. This conjecture proposes a simple one-dimensional relation between Plücker-coordinate data and polytope volume, motivated by the observed one-component multidimensional-scaling projections; the supplied text gives no evidence that the relation has been proved or disproved.