The braid-group preservation conjecture for higher-term quantum Plücker relations

Let ${\mathbb {C}}_q[\operatorname{Gr}(k,n)]$ be the quantum Grassmannian cluster algebra, with quantum Plücker coordinates, and let $\sigma_i$ and $\sigma_i^{-1}$ be the associated braid-group maps. An $r$-term quantum Plücker relation means a quantum Plücker relation having $r$ terms.

Braid-group preservation conjecture. The maps $\sigma_i$ and $\sigma_i^{-1}$ also preserve $r$-term quantum Plücker relations for $r\geq 4$ in ${\mathbb {C}}_q[\operatorname{Gr}(k,n)]$, as well as any other relations in ${\mathbb {C}}_q[\operatorname{Gr}(k,n)]$ that cannot be derived from quantum Plücker relations, if such relations exist.

The authors have proved preservation of the quasi-commutation relations of quantum Plücker coordinates and of the exchange relations, which include the 3-term quantum Plücker relations. It is not known whether the quantum Grassmannian has a presentation by quantum Plücker coordinates modulo all quantum Plücker relations, so the possible additional relations and their preservation remain open.