Spencer's Bus Stop Problem for integral stationary geodesic networks

Fix a Riemannian $n$-manifold $(M,g)$, $n \geq 2$, and let $U \subset\subset V \subset\subset M$ be open. An integral stationary geodesic network is a geodesic network with integer multiplicities that is stationary for the length functional. Spencer's Bus Stop Problem. For any $L \geq 0$, there is a constant $C=C(M,U,V,L)$ such that any integral stationary geodesic network $X \subset M$ whose total length counted with multiplicity in $V$ is at most $L$ has at most $C$ singular points in $U$. This would extend the Euclidean partial $\epsilon$-regularity result to arbitrary Riemannian manifolds and give a uniform local bound on singularities under a local mass bound; the source does not provide evidence that the conjecture has been resolved.