Length-diameter boundedness conjecture for geodesic multinets

Let a geodesic multinet in the Euclidean plane have $n$ unbalanced vertices, diameter $D$, total length $L$, and some number of balanced vertices. Length-diameter boundedness conjecture. There exists a function $f$ such that the number of balanced vertices is less than $f(L/D,n)$. The conjecture is proposed as a sufficient condition for Gromov's conjecture after rescaling the diameter. It is stated for geodesic multinets, and no resolution is given.