Global isoperimetric conjecture for centered circles in the Holmes–Thompson plane

Let $(F_B,B^2(1))$ be the two-dimensional Finsler space form under consideration, and let $c_0$ be the circle centered at the origin. Consider simple closed curves that are smooth and have a fixed length.

Global Holmes–Thompson isoperimetric conjecture. The circle $c_0$ encloses the maximal Holmes–Thompson area among all such curves.

The preceding result establishes that $c_0$ is a proper strong maximum of the isoperimetric problem and that there are no conjugate points along the isoperimetric extremal circles. The conjecture asserts that this local maximality extends to a global maximum.