Orthogonality conjecture for half-space six-vertex symmetric functions

Let $q$, $a$, and $Y=(y_1,y_2,\dots)$ be the parameters of the half-space six-vertex model, and let $F_{\kappa}$ be the associated symmetric function indexed by a finite subset $\kappa$. Fix a finite subset $\nu=\\{\nu_1>\cdots>\nu_n\geq 1\\}$ and a second finite subset $\kappa$ with $|\kappa|\leq n$. Let $\mathcal C$ be a small positively oriented circle surrounding all points $y_j^{-1}$, $j\geq1$, and no other singularities of the integrand, such that $q\cdot\mathcal C$ is disjoint from the interior of $\mathcal C$.

Orthogonality conjecture. In the limit $c\to\infty$,

$$\oint_{\mathcal C}\frac{\mathrm{d}w_1}{2\pi\mathrm{i}}\cdots\oint_{\mathcal C}\frac{\mathrm{d}w_n}{2\pi\mathrm{i}}\prod_{1\leq i<j\leq n}\left[\frac{w_j-w_i}{qw_j-w_i}\frac{1-qw_iw_j}{1-w_iw_j}\right]\prod_{i=1}^{n}\left[\frac{w_i-a}{w_i(1-aw_i)}\frac{1-qw_i^2}{1-w_i^2}\frac{y_{\nu_i}}{1-qw_i y_{\nu_i}}\prod_{j-1}^{\nu_i-1}\frac{1-w_i y_j}{1-qw_i y_j}\right]F_{\kappa}(w_1,\dots,w_n)=\delta_{\kappa,\nu}.$$

This would provide an integral orthogonality relation for the symmetric functions arising from the half-space six-vertex model. The source presents it as conjectural evidence that these functions have orthogonality properties analogous to those of related multivariate rational functions; no resolution is supplied.