Type BC residue expansion identity

Fix $k\geq 1$ and $c\in(0,\infty)$. Let $\alpha_1,\ldots,\alpha_k$ be real numbers and let $F(z_1,\ldots,z_k)$ satisfy $\alpha_k=0$, $\alpha_j>\alpha_{j+1}+c$ for $1\leq j\leq k-1$, and the stated analyticity and decay condition in each variable. For a partition $\lambda=1^{m_1}2^{m_2}\cdots\vdash k$, let $\ell(\lambda)$ denote its length, let $(a)_n$ denote the rising factorial, and define

$$(u_1,\ldots,u_{2\ell(\lambda)})=\big(-w_1+\tfrac{c}{2},w_1-\tfrac{c}{2}+\lambda_1c,\ldots,-w_{\ell(\lambda)}+\tfrac{c}{2},w_{\ell(\lambda)}-\tfrac{c}{2}+\lambda_{\ell(\lambda)}c\big).$$

Also define $E^c$ by symmetrization over the hyperoctahedral group $BC_k$ as in the displayed formula below. Type $BC$ residue expansion conjecture. The following identity should hold:

$$\begin{aligned} &\int_{\alpha_1-\mathbf{i}\infty}^{\alpha_1+\mathbf{i}\infty}\frac{dz_1}{2\pi\mathbf{i}}\cdots\int_{\alpha_k-\mathbf{i}\infty}^{\alpha_k+\mathbf{i}\infty}\frac{dz_k}{2\pi\mathbf{i}}\prod_{1\leq A<B\leq k}\frac{z_A-z_B}{z_A-z_B-c}\frac{z_A+z_B}{z_A+z_B-c}F(\vec z)\\ &=c^k\sum_{\substack{\lambda\vdash k\lambda=1^{m_1}2^{m_2}\cdots}}\frac{(-1)^{\ell(\lambda)}}{m_1!m_2!\cdots}\int_{\alpha_k-\mathbf{i}\infty}^{\alpha_k+\mathbf{i}\infty}\frac{dw_1}{2\pi\mathbf{i}}\cdots\int_{\alpha_k-\mathbf{i}\infty}^{\alpha_k+\mathbf{i}\infty}\frac{dw_{\ell(\lambda)}}{2\pi\mathbf{i}}\\ &\qquad\times\prod_{j=1}^{\ell(\lambda)}\frac{1}{4c}\frac{\left(\frac{2w_j+c}{2c}\right)_{\lambda_j-1}}{\left(\frac{2w_j}{2c}\right)_{\lambda_j}}\operatorname{Pf}\left[\frac{u_i-u_j}{u_i+u_j}\right]_{i,j=1}^{2\ell(\lambda)}E^c(w_1,w_1+c,\ldots,w_{\ell(\lambda)}+c(\lambda_{\ell(\lambda)}-1)). \end{aligned}$$

Here

$$E^c(z_1,\ldots,z_k)=\sum_{\sigma\in BC_k}\prod_{1\leq B<A\leq k}\frac{z_{\sigma(A)}-z_{\sigma(B)}-c}{z_{\sigma(A)}-z_{\sigma(B)}}\frac{z_{\sigma(A)}+z_{\sigma(B)}-c}{z_{\sigma(A)}+z_{\sigma(B)}}F(\sigma(\vec z)).$$

The result would provide the type $BC_k$ analogue of the preceding residue expansion and is intended for the half-space polymer application. The authors state that they have a complete proof modulo a cancellation claim concerning residues generated by contour deformations; partial evidence for that claim is established in the one-string case.