The finite companion hierarchy for Bressoud identities

Let $n,k,i$ be non-negative integers with $k\geq 5$ and $k>i\geq 1$, and set $\alpha_{ij}=\max\{j-i+1,0\}$. The notation ${a\brack b}'_{q^2}$ and ${a\brack b}'_q$ denotes the ordinary $q^2$- and $q$-binomial coefficient, respectively, with the same exceptional convention stated in the source.

Finite companion Bressoud conjecture.

$$\begin{aligned} \sum_{n_1\geq \dots\geq n_{k-1}\geq 0} &q^{n_1^2+\dots+n_{k-1}^2+n_i+\dots+n_{k-1}}{n-\sum_{j=1}^{k-2}n_j-k+i\brack n_{k-1}}'_{q^2}\\ &\quad\times\prod_{j=1}^{k-2}{2n-2\sum_{l=1}^{j-1}n_l-n_j-n_{j+1}-2\alpha_{ij}\brack n_j-n_{j+1}}'_q\\ &=\sum_{r=-\infty}^{\infty}(-1)^r q^{r(kr+k-i)}{2n\brack n-kr+(k-i)\frac{(-1)^r-1}{2}}_q. \end{aligned}$$

This proposes a companion polynomial hierarchy for the Bressoud identities beyond the cases proved in the paper. The cases with $k\leq 4$ are discussed as established computationally, while the stated range remains open.