Baker–Forrester constant-term conjecture for prescribed symmetry

Let $D_1(n_0;n_1;a,b;q)$ be the constant term of the Laurent polynomial

$$\prod_{n_0+1\leq i<j\leq n}(z_i-q^{k+1}z_j)(z_i^{-1}-q^kz_j^{-1})\prod_{i<j}\left(\frac{z_i}{z_j};q\right)_k\left(q\frac{z_j}{z_i};q\right)_k\prod_{i=1}^{n}(z_i;q)_a\left(\frac q{z_i};q\right)_b.$$

Here $\operatorname{CT}$ denotes the constant term, $(x;q)_k$ is the $q$-shifted factorial, and $\Gamma_q$ is the $q$-gamma function. Baker–Forrester constant-term conjecture.

$$\begin{aligned} D_1(n_0;n_1;a,b;q)&=\operatorname{CT}\!\left(\prod_{n_0+1\leq i<j\leq n}(z_i-q^{k+1}z_j)(z_i^{-1}-q^kz_j^{-1})\prod_{i<j}\left(\frac{z_i}{z_j};q\right)_k\left(q\frac{z_j}{z_i};q\right)_k\prod_{i=1}^{n}(z_i;q)_a\left(\frac q{z_i};q\right)_b\right)\\ &=\frac{\Gamma_{q^{k+1}}(n_1+1)}{(\Gamma_q(1+k))^n}\prod_{l=0}^{n_0-1}\frac{\Gamma_q(a+b+1+kl)\Gamma_q(1+k(l+1))}{\Gamma_q(a+1+kl)\Gamma_q(b+1+kl)}\\ &\quad\times\prod_{j=0}^{n_1-1}\frac{\Gamma_q((k+1)j+a+b+kn_0+1)\Gamma_q((k+1)(j+1)+kn_0)}{\Gamma_q((k+1)j+a+kn_0+1)\Gamma_q((k+1)j+b+kn_0+1)}. \end{aligned}$$

The conjecture includes the $n_0=0$ $q$-Morris constant-term identity and has several special cases proved, including $a=b=0$, $n_1=2$, $n_1=3$, and $n\leq5$, but the supplied text does not establish the full statement.