Three-term non-cluster criterion for web invariants

Let $z$ be an indecomposable web invariant and let $z'$ be a cluster variable satisfying a three-term relation

$$zz'=M_1+M_2.$$

Suppose that $M_1$, $M_2$, $z'M_1$, and $z'M_2$ are web invariants, whereas $zM_1$ and $zM_2$ are not.

Three-term non-cluster conjecture. Then $z$ is not a cluster variable.

This is intended as a conjectural test that disqualifies a web invariant from being a cluster variable. The source stresses that the conditions on $z'M_1$ and $z'M_2$ are necessary, but gives no resolution.