Baker–Forrester ratio conjecture for multiblock constant terms

Let $D_p(n_1,\ldots,n_p;n_0;a,b,k;q)$ be the constant term

$$\operatorname{CT}\left(\prod_{\alpha=1}^{p}\prod_{\substack{\min(\widetilde J_\alpha)\leq i<j\leq\max(\widetilde J_\alpha)}}(z_i-q^{k+1}z_j)(z_i^{-1}-q^kz_j^{-1})\prod_{1\leq i<j\leq N}\left(\frac{z_i}{z_j};q\right)_k\left(q\frac{z_j}{z_i};q\right)_k\prod_{i=1}^{n}(z_i;q)_a\left(\frac q{z_i};q\right)_b\right),$$

where $\widetilde J_\alpha$ is determined by the prescribed block sizes. Write $[r]_q!$ for the $q$-factorial and let $n_p>n_j$ for $j=1,\ldots,p-1$. Baker–Forrester ratio conjecture.

$$\frac{D_p(n_1,\ldots,n_{p-1},n_p+1;n_0;0,0,k;q)}{D_p(n_1,\ldots,n_{p-1},n_p;n_0;0,0,k;q)}=\frac{[n_p+1]_{q^{k+1}}}{[k]_q!}\frac{\Gamma_q((k+1)(n_p+1)+k\sum_{j=0}^{p-1}n_j)}{\Gamma_q((k+1)n_p+k\sum_{j=0}^{p-1}n_j)}$$

and equivalently

$$=\frac{[n_p+1]_{q^{k+1}}}{[k]_q!}\frac{[k(n+1)+n_p]_q!}{[kn+n_p]_q!}.$$

This is presented as the second Baker–Forrester conjecture, concerning ratios of multiblock constant terms in the special case $a=b=0$. The supplied text gives no proof or resolution of the general assertion.