Cluster conjecture for inverse-map functions on double Bruhat cells

Let $G$ be a complex reductive group, let $w\in W$ and let $L^{e,w}$ be the corresponding double Bruhat cell. For a decorated double reduced word $(\mathbf{i},K,L)$, write $\mathbb{I}=\mathbb{I}(\mathbf{i},K,L)$, and let $f_1,\ldots,f_n$ be the unique regular functions from Theorem whose Laurent monomials give the factorization parameters of $x\in L^{e,w}$.

Cluster conjecture for inverse-map functions. The set $\{f_1,\ldots,f_n\}$ is a cluster for $L^{e,w}$; equivalently, the fraction field generated by $\{f_1,\ldots,f_n\}$ is the function field of $L^{e,w}$.

The conjecture connects the inverse factorization coordinates with the cluster-algebra structure on $L^{e,w}$. The source does not state whether it has been proved or disproved.