The finite Andrews–Gordon Bailey hierarchy

Let $n,p,k,i$ be integers with $n\geq 0$, $k\geq 4$, and $k>i\geq 1$, and set $\alpha_{ij}=\max\{j-i+1,0\}$.

Finite Andrews–Gordon Bailey hierarchy conjecture.

$$\begin{aligned} &\sum_{m_p\geq\dots\geq m_1\geq0}\frac{q^{m_p^2+\dots+m_1^2}(q;q)_{2n}}{(q;q)_{n-m_p}(q;q)_{m_p-m_{p-1}}\dots(q;q)_{m_2-m_1}(q;q)_{2m_1}}\\ &\quad\times\sum_{n_1\geq\dots\geq n_{k-1}\geq n_k=0}q^{n_1^2+\dots+n_{k-1}^2+n_i+\dots+n_{k-1}}\prod_{j=1}^{k-1}{2m_1-2\sum_{l=1}^{j-1}n_l-n_j-n_{j+1}-2\alpha_{ij}\brack n_j-n_{j+1}}_q\\ &=\sum_{r=-\infty}^{\infty}(-1)^r q^{\frac{r((2k+1)r+2k-2i+1)}{2}+p\left(\frac{(2k+1)r}{2}-(2k-2i+1)\frac{(-1)^r-1}{4}\right)^2}{2n\brack n-\frac{(2k+1)r}{2}+(2k-2i+1)\frac{(-1)^r-1}{4}}_q. \end{aligned}$$

This is obtained by iterating a Bailey-type transformation on the proposed finite companion hierarchy. The cases covered by the paper's direct computations provide evidence, but the general identity remains open.