The graphical-basis conjecture for quantum cluster monomials

Let $\mathsf{BWeb}_{\mathfrak{sl}_{3},{\Sigma}}$ be the graphical basis of the quantum $\mathfrak{sl}_3$ skein algebra of $\Sigma$, and let cluster monomials be monomials formed from cluster variables belonging to a common quantum cluster. The graphical-basis conjecture. The graphical basis $\mathsf{BWeb}_{\mathfrak{sl}_{3},{\Sigma}}$ contains all the cluster monomials. This would place the cluster-monomial elements inside the distinguished graphical basis and strengthen the expected correspondence between quantum clusters and web clusters.