Burago–Ivanov conjecture on strict minimal filling of simple manifolds

A compact Riemannian manifold with boundary is simple when it has strictly convex boundary, every two points are connected by a unique geodesic segment, and it has no conjugate points. It is a strict minimal filling if, whenever a compact Riemannian manifold $M'$ with the same boundary satisfies

$$d_{M'}(x,y)\geq d_M(x,y)\quad\forall x,y\in\partial M,$$

then $\operatorname{Vol}(M')\geq\operatorname{Vol}(M)$, with equality only when $M$ and $M'$ are isometric by an isometry fixing every boundary point.

Burago–Ivanov conjecture. Every simple manifold is a strict minimal filling.

The conjecture concerns volume minimization among Riemannian fillings whose boundary distances dominate those of the given manifold. Its resolution status is not specified in the supplied source context.