The weakly subadditive-labelling criterion for Arnold–Liouville integrability

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

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

for every vertex $z$, without imposing the additional unequal-sums condition. Weakly subadditive-labelling conjecture. The recurrent quiver $Q$ is Arnold-Liouville integrable if and only if it admits a weakly subadditive labelling. The supplied text gives no further context or resolution status for this criterion.