The four-factor $q$-congruence conjecture

Let $n,r,s,t$ be positive integers with $r+s+t\bigl(\equiv 1\pmod 2\bigr)$ and let j\bigl\in\mathbb{N}\bigr). Define $A_{u,k}(q)$ as in the paper, and let

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

or

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

The four-factor $q$-congruence conjecture. The following four congruences hold:

$$[4n+1]\sum_{k=0}^n \eta_k A_{3n,k}(q)^r A_{2n,k}(q)^s A_{n,k}(q)^t\equiv 0\pmod{\dfrac{1}{[6n+1]}{6n+1\brack n}},$$ $$[4n+1]\sum_{k=0}^n \eta_k A_{3n,k}(q)^r A_{2n,k}(q)^s A_{n,k}(q)^t\equiv 0\pmod{\dfrac{1}{[6n+1]}{6n+1\brack 3n}},$$ $$[8n+1]\sum_{k=0}^n \eta_k A_{4n,k}(q)^r A_{2n,k}(q)^s A_{n,k}(q)^t\equiv 0\pmod{{8n+1\brack 3n}},$$ $$[6n+1][8n+1]\sum_{k=0}^n \eta_k A_{4n,k}(q)^r A_{3n,k}(q)^s A_{2n,k}(q)^t\equiv 0\pmod{{8n+1\brack 3n}}.$$

These conjectural congruences generalize the preceding corollaries; the first two are known in related special cases, but the stated general assertions remain open.