The u-refined Cauchy identity for lifted Koornwinder polynomials

Let $m_0(\lambda)$ be the multiplicity of zero parts of the partition $\lambda$, let $P_\lambda(x_1,\dots,x_n;t)$ be the Hall–Littlewood polynomial, and let $b_\lambda(t)$ be its standard coefficient. Write $\bar y_j=y_j^{-1}$, and let $\tilde K_\lambda$ be the lifted Koornwinder polynomial with parameters $(q,t,T;t_0,t_1,t_2,t_3)=(0,t,ut^{n-1};t_0,t_1,t_2,t_3)$. The $u$-refined lifted Koornwinder Cauchy conjecture.

$$\begin{aligned} &\sum_{\lambda} \prod_{i=1}^{m_0(\lambda)} (1-u t^{i-1}) b_{\lambda}(t) P_{\lambda}(x_1,\dots,x_n;t) \tilde{K}_{\lambda}(y_1^{\pm 1},\dots,y_n^{\pm 1}; 0, t, u t^{n-1}; t_0, t_1, t_2, t_3) \\ &= \prod_{i=1}^{n} \frac{(1- t_0 x_i )(1-t_1 x_i)(1-t_2 x_i)(1-t_3 x_i)}{(1-t x_i^2)} \frac{\prod_{i,j=1}^{n} (1- t x_i y_j) (1- t x_i \bar y_j)} {\prod_{1 \leqslant i<j \leqslant n} (x_i-x_j) (y_i-y_j) (1 - t x_i x_j) (1-\bar y_i \bar y_j)} \\ &\qquad\times \det_{1\leqslant i,j \leqslant n} \left[ \frac{1-u + (u-t) (x_i y_j + x_i \bar y_j) + (t^2 - u) x_i^2} {(1-x_i y_j) (1-t x_i y_j)(1-x_i \bar y_j) (1-t x_i \bar y_j)} \right]. \end{aligned}$$

This is presented as a refinement of a Cauchy identity for lifted Koornwinder polynomials. The notation and relevant background are deferred to Section 7 and an appendix, and the source does not provide a proof or a resolution.