The subadditive-labelling criterion for Zamolodchikov integrability

Let $Q$ be a recurrent quiver: its underlying graph is bipartite, and mutating at its black vertices followed by mutating at its white vertices returns the same quiver. A positive labelling $\nu\colon Q\to\mathbb R_{>0}$ is subadditive if, for every vertex $z$,

$$\nu(z)\geq\frac12\max\left(\sum_{y\to z}\nu(y),\sum_{z\to y}\nu(y)\right),$$

and equality can occur only when the incoming and outgoing sums are unequal. Subadditive-labelling conjecture. The recurrent quiver $Q$ is Zamolodchikov integrable if and only if it admits a subadditive labelling. This would give a combinatorial criterion for integrability beyond box products; its resolution status is not specified in the supplied text.