Equidistribution conjecture for stationary geodesic nets

Let $M$ be a closed manifold, let $g$ be a Riemannian metric on $M$, and let $\gamma_i:\Gamma_i\rightarrow M$ be stationary geodesic nets. A sequence of such nets is equidistributed in $M$ if their length-weighted integrals converge to the normalized Riemannian volume integral. Equidistribution conjecture. For a generic set of metrics, every metric $g$ in that set admits a sequence $\{\gamma_i:\Gamma_i\rightarrow M\}$ of stationary geodesic nets in $(M,g)$ such that, for every $C^{\infty}$ function $f:M\rightarrow\mathbb{R}$,

$$\lim_{k\rightarrow\infty}\frac{\sum_{i=1}^k\int_{\gamma_i}f\,\operatorname{dL}_g}{\sum_{i=1}^k\operatorname{L}_g(\gamma_i)}=\frac{\int_M f\,\operatorname{dVol}_g}{\operatorname{Vol}(M,g)}.$$

This is an open problem in general; the cases $n=2$ and $n=3$ were stated in the source to be solved, with the two-dimensional case established for closed geodesics and generic metrics.