The bijection between alcove covers and hypersimplicial decorated ordered set partitions

Let $n$ be any positive integer and let $A_0$ be an alcove in the hypersimplex $\Delta_{2,n}$. Let $\mathcal{P}_{A_0}=(V,E)$ be the breadth-first search order of $\Gamma_{2,n}$ rooted at $A_0$. For $v\in V$, define

$$\operatorname{cover}(v)=\#\{u\in V\mid u\prec\cdot v\text{ in }\mathcal{P}_{A_0}\}$$

to be the number of elements covered by $v$. Bijection conjecture. The map $\psi$ is a bijection between

$$\{v\in V\mid \operatorname{cover}(v)=d\}$$

and the set of hypersimplicial decorated ordered set partitions of type $(2,n)$ with winding number $d$. This would identify the breadth-first search structure of the alcove graph with the enumerative structure of hypersimplicial decorated ordered set partitions; the source does not provide resolution, so the claim remains open.