Pitman–Stanley flow-polytope volume conjecture for the net flow vector $(1,0,1,0,\ldots,0)$

Let $\textup{PS}_{n+1}$ be the Pitman–Stanley graph on $n+1$ vertices, and let $\mathcal{F}_{\textup{PS}_{n+1}}(1,0,1,0,\ldots,0)$ denote its flow polytope with the indicated net flow vector. Pitman–Stanley volume conjecture. For $n\geq 2$,

$$\operatorname{vol} \mathcal{F}_{\textup{PS}_{n+1}}(1,0,1,0,\ldots,0)=2^{n-1}-n-2.$$

The formula is proposed as a consequence of techniques used for the preceding volume computation, but no resolution is given here.