Varchenko's path-sum conjecture for stable-envelope integrals

Let $X=T^*Gr(k,n)$, let $\boldsymbol{a}=(a_1,\ldots,a_n)$ be the equivariant parameters, and let $\boldsymbol{a}\to 0$ denote the non-equivariant limit. For a partition $\lambda$ in an $(n-k)\times k$ rectangle, let $\lambda^t$ be its conjugate and set $\Omega(\lambda)=\lambda^t+(k,k-1,\ldots,1)$. Write $p_{\Omega(\lambda)}$ for the corresponding torus-fixed point, and let $\mathbb{V}(\lambda)$ be the weighted path sum defined in the paper. Varchenko's conjecture.

$$\mathbb{V}(\lambda)=\sum_{p_J\in X^{\mathsf{T}}}\frac{\operatorname{Stab}(p_{\Omega(\lambda)})|_{p_J}}{e(T_{p_J}X)}.$$

Consequently,

$$\lim_{\boldsymbol{a}\to 0}\mathbb{V}(\lambda)=\int_X\operatorname{Stab}(p_{\Omega(\lambda)}).$$

This proposes a combinatorial path-sum formula for equivariant stable-envelope integrals and their non-equivariant limits; the source presents it as a conjectural method due to A. Varchenko.