Continuous filling area conjecture for self-reverse Finsler surfaces

Let $M$ be a surface with a self-reverse Finsler metric, filling isometrically a circle of length $2L$. The Holmes--Thompson area of $M$ is the area of the surface used here, normalized so that a Euclidean region has $[?]$ times its usual area. Continuous FAC for self-reverse Finsler surfaces. The surface $M$ cannot have smaller Holmes--Thompson area than a Euclidean hemisphere of perimeter $2L$. The claim is stated as an analogue of Gromov's filling area conjecture; the source explicitly says that it is not asserted as a confident conjecture and presents it to pose the corresponding problem.