Three-term exchange-relation criterion for web invariants

Let $z$ and $z'$ be indecomposable web invariants satisfying the three-term relation

$$zz'=M_1+M_2.$$

Assume that $M_1$, $M_2$, $zM_1$, $zM_2$, $z'M_1$, $z'M_2$, and $M_1M_2$ are web invariants.

Three-term cluster conjecture. Then $z$ and $z'$ are cluster variables, and the displayed relation is an exchange relation.

This conjecture aims to identify cluster variables and exchange relations from multiplicative web-basis conditions around a three-term relation. The source notes that the expected exchange relations should satisfy these conditions, but gives no resolution.