The quantum T-system conjecture for quantum minors

Let $\mathbf{i}=(i_1,\ldots,i_l)$ be a reduced expression of a braid group element $b\in\operatorname{Br}^+$, and let $[a,c]\subset[1,l]$ be an $i$-box, meaning that $i_a=i_c$. For each such box, let $\widehat{D}_{\mathbf{i}}[a,c]$ denote the corresponding quantum minor, defined as the global basis element associated with the sequence supported on positions in $[a,c]$ whose letter is $i_a$. For an index $r$, write $r^+$ and $r^-$ for the relevant neighboring occurrences of the same simple index, and write $a^+(j)$ and $c^-(j)$ for the corresponding neighboring positions for $j$.

Quantum T-system conjecture. For any $i$-box $[a,c]$, there exist constants $A$ and $B$ such that

$$\widehat{D}_{\mathbf{i}}[a^+,c]\widehat{D}_{\mathbf{i}}[a,c^-]=q^A\widehat{D}_{\mathbf{i}}[a,c]\widehat{D}_{\mathbf{i}}[a^+,c^-]+q^B\prod_{d(i_a,j)=1}\widehat{D}_{\mathbf{i}}[a^+(j),c^-(j)].$$

The relation is a quantum-minor analogue of a T-system. It is known to hold in Dynkin cases by Kashiwara, Kim, Oh, and Park, but the supplied text does not establish it in the generality stated.