Conjectural duality for iterated $q$-integrals on ${\mathbb{P}}^1$ minus four generic points

We propose a conjectural $q$-analogue of the classical duality for iterated integrals on ${\mathbb{P}}^1$ minus four points, arising from the involutive Möbius transformation which exchanges the four marked points in pairs. To this end, we introduce iterated $q$-integrals with position-dependent $q$-shifts of the parameters and define a functional on admissible words in the six pairwise letters. The conjecture states that this functional is invariant under a natural anti-automorphism of the word algebra. We relate the conjecture to Yamamoto’s duality for one-variable multiple $q$-polylogarithms. Finally, we prove the conjecture in several special cases.