The all-$j$ conjecture for the two alternating $q$-congruence theorems

Let the two preceding results, Theorems $\mathrm{thm:fatorodd\text{-}first}$ and $\mathrm{thm:fatorodd\text{-}second}$, be the stated congruence theorems involving the parameter $j$. All-$j$ conjecture. Theorems $\mathrm{thm:fatorodd\text{-}first}$ and $\mathrm{thm:fatorodd\text{-}second}$ hold for every $j\in\mathbb{N}$. The conjecture asserts extension of those theorems to all nonnegative values of $j$; the supplied text gives no resolution.