Convex geodesic Swiss cheese conjecture

Let $(N,g)$ be a compact smooth Riemannian manifold. A finite disjoint union of smooth convex balls in $N$ is a collection of smooth balls that are convex with respect to the metric $g$; let $M$ be their complement.

Convex geodesic Swiss cheese conjecture. There exists such a finite disjoint union whose removal from $N$ delivers a metric $g_M$ of gradient type on $M$.

The preceding result establishes the analogous statement without requiring the balls to be convex in the original metric, so the additional convexity requirement is left as a conjectural strengthening.