Gromov's boundedness conjecture for geodesic nets

Let a geodesic net in the Euclidean plane have a finite number of unbalanced vertices, each with an imbalance, and let the total imbalance be the sum of those imbalances. Gromov's conjecture. The number of balanced vertices of a geodesic net in the Euclidean plane can be bounded above in terms of the number of unbalanced vertices and the total imbalance. This would provide a quantitative bound despite the absence of a bound depending only on the number of unbalanced vertices. A related result is stated for geodesic nets whose imbalances are all equal to one, but the general conjecture remains open.