Gromov's bounded-imbalance conjecture for planar geodesic nets

Let $G=(B,U,E)$ be a geodesic net in the plane, with balanced vertices $B$ and unbalanced vertices $U$. For a vertex $v$, let $\operatorname{imb}(v)$ denote the norm of the sum of the unit tangent vectors to the incident edges. For $n\in\mathbb{N}$ and $c\in\mathbb{R}$, the total imbalance is $\sum_{v\in U}\operatorname{imb}(v)$. Gromov's conjecture. There is a function

$$g:\mathbb{N}\times\mathbb{R}\to\mathbb{N}$$

such that every geodesic net $G=(B,U,E)$ in the plane with $|U|\leq n$ and

$$\sum_{v\in U}\operatorname{imb}(v)\leq c$$

has $|B|\leq g(n,c)$. This is presented as an equivalent formulation of Gromov's question about bounding balanced vertices. The paper does not establish the bound in general, so the conjecture remains open.