The u-deformed Macdonald Littlewood identity

Throughout this subsection, let $N=2n$, and let $P_\lambda(x_1,\dots,x_N;q,t)$ denote Macdonald polynomials, with $b^{\rm el}_\lambda(q,t)$ the even-columns Littlewood coefficient. Write $\lambda'$ even when every column length of the partition $\lambda$ is even. Let $\operatorname{Pf}$ denote the Pfaffian of the displayed skew-symmetric matrix. The $u$-deformed Macdonald Littlewood conjecture.

$$\sum_{\lambda'\ {\rm even}} \ \prod_{i\ {\rm even}}^{N} (1-u q^{\lambda_i} t^{N-i} ) b^{\rm el}_{\lambda}(q,t) P_{\lambda}(x_1,\dots,x_{N};q,t) = \prod_{1 \leqslant i<j \leqslant N} \frac{(t x_i x_j;q)}{(x_i x_j;q)} \prod_{1 \leqslant i<j \leqslant N} \frac{(1-x_i x_j)}{(x_i - x_j)} \operatorname{Pf}_{1\leqslant i < j \leqslant N} \left[ \frac{(x_i - x_j) (1 - u + (u-t) x_i x_j)} {(1-x_i x_j) (1-t x_i x_j)} \right].$$

This conjecture is a one-parameter refinement of the even-columns Littlewood identity for Macdonald polynomials. It was supported by extensive numerical testing, but the source explicitly states that no proof is known; a possible approach is via suitable difference operators.