Boundedness conjecture for geodesic nets on Riemannian manifolds

Let $M$ be a complete Riemannian manifold. For a geodesic net $G$ on $M$, let $L$ be its total length, $\widetilde L$ its adjusted length, $D$ its diameter, and $n$ the number of unbalanced vertices. The adjusted length is the sum over edges $e_i$, parametrized by arclength, of the integrals of $1/\operatorname{inj}(e_i(s))$, where $\operatorname{inj}$ is the injectivity radius of $M$. Boundedness conjecture for geodesic nets on Riemannian manifolds. There exists a function $f_M$, depending on $M$ but invariant under rescaling $M$, such that the number of balanced vertices of $G$ does not exceed

$$f_M\left(\widetilde L,\frac{L}{D},n\right).$$

In particular, if $M$ has injectivity radius $\operatorname{inj}>0$, then the number of balanced vertices does not exceed

$$f_M\left(\frac{L}{\operatorname{inj}},\frac{L}{D},n\right).$$

The adjustment accounts for collapsing injectivity radius, which gives counterexamples to unadjusted bounds on general manifolds. The conjecture remains open.