The volume formula for interval-vector polytopes

Let $n$ and $i$ be parameters for the interval-vector polytope $\mathcal{P}_n(\mathcal{I}_{1,n-i})$, where $\mathcal{I}_{1,n-i}$ denotes the corresponding family of intervals and $\operatorname{vol}$ denotes volume. Volume conjecture.

$$\operatorname{vol}(\mathcal{P}_n(\mathcal{I}_{1,n-i}))=2^{i}(n-(i+1)).$$

This formula was observed from computations of $f$-vectors generated with polymake; the supplied text gives no proof or resolution, so the conjecture remains open.