Cluster-variable recognition conjecture for KR-polynomials

Let $s\in\mathbb{Z}$, let $\mathfrak{S}_s$ be the indicated quantum seed, and let ${}^{s}\mathfrak{u}_{i,p}$ denote its seed variables. Let $F_q(\underline{m}^{(j)}[a,b])\in\mathfrak{K}_q(\mathsf{g})$. Cluster-variable recognition conjecture for KR-polynomials. If $F_q(\underline{m}^{(j)}[a,b])$ $q$-commutes with every ${}^{s}\mathfrak{u}_{i,p}\in\mathfrak{S}_s$, then there exists $(j,l)$ in the indicated folded Dynkin diagram such that

$${}^{s}\mathfrak{u}_{j,l}=F_q(\underline{m}^{(j)}[a,b]).$$

The claim characterizes these KR-polynomials among elements commuting with all variables of the seed; no resolution status is supplied.