Duality symmetry of the confluence expansion

Prove the conjectured membership and duality identities for the confluence expansion $\varphi^{\pi}(\theta(\widetilde{\boldsymbol{k}}))$ of every admissible augmented index, namely that it belongs to $V_{\mathrm{BZ}}\otimes\mathbb{Q}\langle\mathcal{B}\rangle$ and is transformed into the corresponding expansion for the augmented dual by $\tau_{\mathrm{BZ}}\otimes\mathrm{id}$.

Background

The paper defines theta-values from admissible augmented indices and applies the confluence expansion map φπ\varphi^{\pi}. Numerical computations suggest that the first tensor factor lies in the subspace generated by BradleyZhao elements and that the expansion is compatible with augmented-index duality.

These identities are used later to derive the inclusion of BradleyZhao and SchlesingerZudilin duality relations in the confluence relations. They remain unresolved in the paper and are presented as its main conjecture in this section.

References

We now state the main conjecture of this section, which is suggested by numerical computations. For $\widetilde{\boldsymbol{k}}\in\operatorname{Ind}{\mathrm{aug}}$, we have $$\varphi{\pi}(\theta(\widetilde{\boldsymbol{k}}))\in V{\mathrm{BZ}}\otimes\mathbb{Q}\langle\mathcal{B}\rangle.$$ Furthermore, for $\widetilde{\boldsymbol{k}}\in\operatorname{Ind}{\mathrm{aug}}$, we have $$\varphi{\pi}(\theta(\widetilde{\boldsymbol{k}}))=(\tau{\mathrm{BZ}}\otimes\mathrm{id})(\varphi{\pi}(\theta(\widetilde{\boldsymbol{k}}{\dagger}))).$$

— Confluence relations for $q$-analogues of multiple zeta values  (2609.01458 - Hirose, 1 Sep 2026) in Conjecture 4.5, Section 4.2, “A conjectural formula”

The duality conjecture. It remains to prove \Cref{conj:phi_dual}.

— Confluence relations for $q$-analogues of multiple zeta values  (2609.01458 - Hirose, 1 Sep 2026) in Section 5, “Further questions”