Conjectural duality for iterated q-integrals on the projective line minus four points

Let $N\geq 0$, let $K$ be the field of fractions of $\mathbb{Q}[A,B,C,D,q]/(A-q^{N}D)$, and let $W^{0}$ be the space containing words $(u_{1}v_{1})\cdots(u_{k}v_{k})$ with letters in $\{A,B,C,D\}$. For $u_{j},v_{j}\in\{A,B,C,D\}$, define

$$\begin{aligned} A^{(j)}&:=Aq^{\#\{h\leq j\mid u_{h}v_{h}\in\{BC,BD,CD\}\}},\\ B^{(j)}&:=Bq^{\#\{h\leq j\mid u_{h}v_{h}\notin\{AC,AD\}\}+\#\{h\geq j\mid u_{h}v_{h}=CD\}},\\ C^{(j)}&:=Cq^{\#\{h\leq j\mid u_{h}v_{h}\notin\{AB,AD\}\}+\#\{h\geq j\mid u_{h}v_{h}=BD\}},\\ D^{(j)}&:=Dq^{-\#\{h\geq j\mid u_{h}v_{h}\in\{AB,AC,BC\}\}}. \end{aligned}$$

Define $L_q:W^0\to K$ by

$$L_q((u_1v_1)\cdots(u_kv_k)):=I_q(A;[u_1^{(1)},v_1^{(1)}],\dots,[u_k^{(k)},v_k^{(k)}];D).$$

Here $\tau$ denotes the word-reversing involution from the preceding classical duality theorem. Conjectural duality. For every $w\in W^{0}$,

$$L_q(w)=L_q(\tau(w)).$$

This conjecture proposes that the duality for iterated integrals on the projective line minus four points persists for the corresponding iterated $q$-integrals under the relation $A=q^ND$. Its classical limit recovers the real-variable duality, but the conjecture is presented as the main unproved assertion of the paper; no resolution is supplied in the given text.