Filling minimality of Finslerian 2-discs

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this result for Finsler metrics with area defined as the two-dimensional Holmes–Thompson volume. This implies a generalization of Pu's isosystolic inequality to Finsler metrics, both for the Holmes–Thompson and Busemann definitions of the Finsler area.