$G$. For an excellent introduction to this conjecture we refer the reader to Goodearl’s paper [@Goo1]. The conjecture appears extremely difficult to prove and it is only known to be true in the cases of $G=SL_2, SL_3$.

The coordinate rings $\CC[G]$ are known to have an upper cluster algebra structure ([@BFZ]) while the quantized coordinate rings are conjectured to have a quantum (upper) cluster algebra structure ([@bz-qclust Conjecture 10.10]). Indeed, it follows from recent results of Geiß, Leclerc and Schröer ([@GLSq Section 12.4]) that $\CC_q[SL_n]$ has a quantum cluster algebra structure. Cluster algebras are nowadays very well-established, hence we do not recall any of the definitions here, and refer the reader to the literature, resp. our Section \[se:Cluster Algebras\]. Most importantly for our purposes, a cluster algebra over $\CC$ is defined by a combinatorial datum in a field of fractions $\CC(x_1,\ldots, x_n)$. We will denote this [*initial seed*]{} by $({\bf x}, B)$ where ${\bf x}=(x_1,\ldots,x_n)$ and $B$ is an integer $m\times n$-matrix with $m\le n$ such that its principal $m\times m$ submatrix is skew-symmetrizable. The cluster variables $x_{m+1},\ldots, x_n$ are the frozen variables which we will call coefficients. A quantum cluster algebra is given by a [*quantum seed*]{} $( {\bf 