Macdonald polynomial vanishing conjecture for the $U(2n)$ Grassmannian coset

Let $n\geq 0$, let $\mu\overline{\nu}$ be a dominant weight of $U(2n)$, and let $Z$ normalize the integral so that the trivial polynomial integrates to $1$. $U(2n)$ Grassmannian conjecture.

$$\frac{1}{Z}\int P_{\mu\overline{\nu}}(\pm\sqrt{x_1},\pm\sqrt{x_2},\dots,\pm\sqrt{x_n};q,t)\prod_{1\leq i\ne j\leq n}\frac{(x_i/x_j;q^2)}{(t^2x_i/x_j;q^2)}\prod_{i=1}^n\frac{dx_i}{2\pi\sqrt{-1}x_i}=0$$

unless $\mu=\nu$; in that case it equals

$$\frac{(-1)^{|\mu|}C^-_\mu(q;q,t)C^+_\mu(t^{2n-2}q;q,t)C^0_\mu(t^n,-t^n;q,t)}{C^-_\mu(t;q,t)C^+_\mu(t^{2n-2}t;q,t)C^0_\mu(qt^{n-1},-qt^{n-1};q,t)}.$$

The conjecture is motivated by integration over a coset of $U(n)\times U(n)$ and was inferred from low-rank cases; the general formula remains open.