Lipschitz-Volume rigidity of manifolds among integral current spaces

Must a surjective 1-Lipschitz map between spaces of equal volume necessarily be an isometric homeomorphism? (This is referred to as Lipschitz-volume rigidity.) The answer is yes for closed manifolds and, under additional assumptions, for manifolds with boundary but no in general. Still, Lipschitz-volume rigidity holds in more generality: in joint work with Basso and Creutz we showed rigidity when the domain is an integral current space and the target a convex body in Euclidean space. Recently, Zust completed the picture by showing that rigidity holds also when the domain is a Riemannian manifold. In this talk I will explain the main steps in his proof.