Forest-diagram characterization of cluster monomials

Let $z$ be a web invariant represented by a tensor diagram, and let $D$ be a forest diagram, meaning that the unclasping of $D$ has no cycles.

Forest-presentation conjecture. The web invariant $z$ is a cluster monomial if and only if $z=[D]$ for some forest diagram $D$.

This conjecture extends the proposed tree characterization from individual cluster variables to cluster monomials. It is part of the arborization program and is not resolved in the source.