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An exact differential two-form in the injective hull of the Riemannian circleis constructed and its norm is shown to be stationary at points of the openhemisphere spanned by the circle. The norm employed is the comass with respectto the inscribed Riemannian definition of area on normed planes. This impliesthat in any metric space, the induced Finsler mass of a two-dimensionalAmbrosio-Kirchheim rectifiable metric current with a Riemannian circle oflength $2\pi$ as a boundary can be estimated from below by $2\pi$ plus a termof second order in the Hausdorff distance to an isometric copy of thehemisphere with the same boundary. These competitors contain the class oforiented Lipschitz surfaces of arbitrary topological type and as such theresult provides positive evidence for Gromov's filling area conjecture.

The Riemannian hemisphere is almost calibrated in the injective hull of its boundary