The geodesic Swiss cheese existence conjecture

Let $(N,g)$ be a compact Riemannian manifold. A geodesic Swiss cheese model is the complement $M_{\mathcal B}=N\setminus\bigl(\bigcup_\alpha\operatorname{int}(B_\alpha)\bigr)$ of a finite collection of disjoint closed smooth balls, each geodesically strictly convex, such that every connected component of the intersection of any geodesic with $M_{\mathcal B}$ is a closed segment or a singleton. Geodesic Swiss cheese conjecture. Every compact Riemannian manifold $(N,g)$ admits a geodesic Swiss cheese model. The claim asserts the existence of a non-trapping domain obtained by removing finitely many convex balls; the source gives no resolution.