The general $q$-congruence conjecture for products of $A_{n+i-1,k}(q)$

Let n,r_1,\ldots,r_m\bigl\in\mathbb{Z}^{+}\bigr) with r_1+\cdots+r_m\bigl\equiv 1\pmod 2\bigr) and let $j\in\mathbb{N}$. Define $A_{u,k}(q)$ as in the paper, and let

$$\eta_k=q^{j(k^2+k)}$$

or

$$\eta_k=(-1)^kq^{\binom{k+1}{2}+j(k^2+k)}.$$

General $q$-congruence conjecture. One has

$$\sum_{k=0}^n \eta_k\prod_{i=1}^m A_{n+i-1,k}(q)^{r_i}\equiv 0\pmod{\dfrac{1}{[n+1]}{2n\brack n}}.$$

This extends the preceding corollary. It is proved for $m=1$ in the stated range and for $m=2$ in the stated range; the $q=1$ case has been checked for several additional values, while the full assertion remains open.