Macdonald polynomial vanishing conjecture for a nonstandard $U(2n)$ weight

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 $1$. Nonstandard $U(2n)$ vanishing conjecture.

$$\frac{1}{Z}\int P_{\mu\overline{\nu}}(x_1,\dots,x_n,y_1,\dots,y_n;q,t)\prod_{1\leq i,j\leq n}\frac{(qx_i/y_j,qy_i/x_j;q^2)}{(tx_i/y_j,ty_i/x_j;q^2)}\prod_{1\leq i\ne j\leq n}\frac{(x_i/x_j,y_i/y_j;q^2)}{(qtx_i/x_j,qty_i/y_j;q^2)}\prod_{i=1}^n\frac{dx_i}{2\pi\sqrt{-1}x_i}=0$$

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

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

The weight is explicitly noted to be nonstandard rather than Macdonald or Koornwinder type; the associated orthogonal polynomials are suggested as an interesting object, and the assertion remains conjectural.