Web-combinatorics conjecture for the cluster structure of Gr(3,n)

Let $\mathbb{C}[\operatorname{Gr}^{\sim}(3,n)]$ be the coordinate ring of the affine cone over the Grassmannian. A web invariant is called indecomposable, non-elliptic, or arborizable according to the tensor-diagram properties defined in the paper. Web-combinatorics conjecture. In the cluster structure for $\mathbb{C}[\operatorname{Gr}^{\sim}(3,n)]$: (1) the set of cluster and frozen variables coincides with the set of indecomposable, non-elliptic, arborizable web invariants; (2) two such cluster variables lie in the same cluster if and only if their product is again a non-elliptic web invariant; and (3) if $n\geq 9$, there are infinitely many non-elliptic webs that are both indecomposable and non-arborizable. This predicts both the cluster variables and compatibility in terms of tensor-web combinatorics, while allowing infinitely many non-cluster web invariants for $n\geq 9$.