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