Bivariant web-and-forest and compatibility conjecture

Let $S$ be a marked bordered surface, and consider an $\operatorname{SL}_3$ Fock--Goncharov cluster algebra associated with $S$. A bi-web invariant is the invariant associated with an irreducible bivariant tensor diagram. A forest bivariant tensor diagram is a bivariant tensor diagram whose unclasping has no cycles.

Bivariant web conjecture. In any such cluster algebra: all cluster variables are bi-web invariants; a bi-web invariant is a cluster monomial if and only if it can also be written as a forest bivariant tensor diagram; and compatibility of cluster/coefficient variables is equivalent to their product being a single bi-web invariant.

This extends the preceding web and forest conjectures from ordinary webs to the bivariant Fock--Goncharov setting. The supplied text gives no resolution status.