Conjecture that rank 2 webs give all rank 2 cluster variables

Let $\mathcal{W}$ be the set of rank $2$ webs $W(R,S,T,V)$ on the Grassmannian $\operatorname{Gr}(k,n)$, where $R,S,T$ are cyclically separated subsets and $V$ is disjoint from them, with $|R|+|S|+|T|+2|V|=2k$. Rank 2 web completeness conjecture. The set $\mathcal{W}$ is the complete set of rank $2$ cluster variables on the Grassmannian $\operatorname{Gr}(k,n)$. The conjecture would identify the explicitly constructed rank $2$ webs with all rank $2$ cluster variables; the paper provides evidence for it.