The strict minimal filling conjecture for simple manifolds

Let $M$ be a compact Riemannian manifold with boundary. It is a strict minimal filling if, for every compact Riemannian manifold $M'$ with $\partial M'=\partial M$, the inequalities

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

imply

$$\operatorname{vol}(M')\geq\operatorname{vol}(M),$$

and equality implies that $M$ and $M'$ are isometric by an isometry identical on the boundary. A manifold is simple if its boundary is strictly convex, every two points are joined by a unique geodesic, and it has no conjugate points. Strict minimal filling conjecture. Every simple manifold is a strict minimal filling. Minimal filling is the volume-minimization formulation of boundary rigidity, so this conjecture would give both the volume inequality and rigidity in the equality case for simple manifolds. Its resolution status is not specified in the supplied text.