The case of simply-connected fillings is equivalent to Pu's inequality. Recently the case of genus-1 fillings was settled affirmatively, as well (see reference by Bangert et al. below). Namely, it turns out that one can exploit a half-century old formula by J. Hersch from integral geometry. Namely, consider the family of figure-8 loops on a football, with the self-intersection point at the equator (see figure at the beginning of the article). Hersch's formula expresses the area of a metric in the conformal class of the football, as an average of the energies of the figure-8 loops from the family. An application of Hersch's formula to the hyperelliptic quotient of the Riemann surface proves the filling area conjecture in this case.