Web-basis conjecture for cluster monomials

Let $R_\sigma(V)$ be an invariant ring with extended clusters, cluster monomials, and web invariants. A cluster monomial is a monomial in the elements of one extended cluster.

Cluster-monomial web-basis conjecture. All cluster monomials are web invariants.

This would place all cluster monomials inside the distinguished web basis and is presented as a broad conjectural property of the invariant rings. The source gives no resolution.