Generic unboundedness of balanced vertices for planar geodesic nets

Let $N_0$ be a positive integer and let $S$ be an $N_0$-tuple of points in $\bigl(\mathbb{R}^2\bigr)^{N_0}$. Generic unboundedness conjecture. There exist $N_0$ and an $N_0$-tuple $S$ such that, for every $N$, there is a geodesic net whose set of unbalanced vertices is $S$ and whose number of balanced vertices is greater than $N$. Moreover, the set of such $N_0$-tuples contains a subset of positive measure, or even a non-empty open subset, of $\bigl(\mathbb{R}^2\bigr)^{N_0}$. The conjecture is motivated by constructions with fourteen unbalanced vertices and arbitrarily many balanced vertices; the authors note that $N_0=4$ is possible.