  We find maximal representatives within equivalence classes of metric spheres.
For Ahlfors regular spheres these are uniquely characterized by satisfying the
seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume
rigidity. We also apply our construction to solutions of the Plateau problem in
metric spaces and obtain a variant of the associated intrinsic disc studied by
Lytchak--Wenger, which satisfies a related maximality condition.
