Chow's basis conjecture for the equivariant cohomology of the permutohedral variety

Let $\mathcal H_n$ be the permutohedral variety, let $\mathcal G_k$ be the indexed family of elements defined in the source, let $\widehat{\sigma}_w\in H_T^*(\mathcal H_n)$ be the associated equivariant cohomology class for $w\in\mathcal G_k$, and let $\mathfrak S_w$ be its stabilizer subgroup in $\mathfrak S_n$. For $v\in\mathfrak S_n$, write $\bar v$ for the coset containing $v$ in $\mathfrak S_n/\mathfrak S_w$.

Erasing marks conjecture. The collection

$$\bigcup_{k=0}^{n-1}\bigcup_{w \in \mathcal G_k}\{ v \cdot \widehat{\sigma}_w \mid \bar{v} \in \mathfrak S_n/\mathfrak S_w\}$$

forms a basis of the equivariant cohomology space $H_T^*(\mathcal H_n)$.

The conjecture is attributed to Chow, and the source states that its proof follows from the main theorem in the paper; it is therefore solved by the cited result. The supplied context does not fully define the indexing sets $\mathcal G_k$ or all constructions entering $\widehat{\sigma}_w$.