Isomorphism conjecture for the q-deformed multiple-zeta-value and quantum Grothendieck–Teichmüller representations

Let $\mathcal{Z}_q$ be the Hopf algebra of $q$-deformed multiple polylogarithms, let $\widetilde{\mathcal{H}_{GT}}$ be the Hopf algebra introduced as a quantum analogue of the Grothendieck–Teichmüller group, and let $S^{4,\theta}$ be the noncommutative four-sphere on whose quantum field theories these algebras act. Consider the corresponding representations obtained from these actions.

Representation isomorphism conjecture. The corresponding representations of $\mathcal{Z}_q$ and $\widetilde{\mathcal{H}_{GT}}$ on quantum field theories on $S^{4,\theta}$ are isomorphic.

This conjecture proposes that the $q$-deformed multiple-zeta-value Hopf algebra and the quantum Grothendieck–Teichmüller Hopf algebra induce the same representation-theoretic structure in quantum field theory. The source provides no resolution, so the conjecture remains open.