Finite q-duality conjecture for words in three letters

Let $\mathfrak{h}:=\mathbb{Q}\langle x,y,z\rangle$ and set $\mathfrak{h}^{0}:=\mathbb{Q}\oplus y\mathfrak{h}x$. Fix $N\geq0$. Define $Z_{N,q}:\mathfrak{h}^{0}\to\mathbb{Q}(q)$ by

$$Z_{N,q}(u_{1}\cdots u_{k}):=\sum_{0\leq n_{1}\leq\cdots\leq n_{k}\leq N}\prod_{\substack{1\leq j\leq k\\ u_{j}=x }}\frac{1}{1-q^{n_{j}+\#\{h\leq j\mid u_{h}=y\}}}\prod_{\substack{1\leq j\leq k\\ u_{j}=y }}\frac{-1}{1-q^{n_{j}-N-\#\{h\geq j\mid u_{h}=x\}}}$$ $$\qquad\qquad\times\prod_{\substack{1\leq j\leq k\\ u_{j}=z }}\left(\frac{1}{1-q^{n_{j}+\#\{h\leq j\mid u_{h}=y\}}}-\frac{1}{1-q^{n_{j}-N-\#\{h\geq j\mid u_{h}=x\}}}\right).$$

Finite q-duality conjecture. For every word $w=u_{1}\cdots u_{k}\in\mathfrak{h}^{0}$,

$$Z_{N,q^{-1}}(w)=(-1)^{k}Z_{N,q}(w).$$

This is the specialization of the paper's iterated $q$-integral duality when $B=C=\infty$. The claim remains conjectural in the supplied text, although the paper proves it for some special families of words.