Minimal area of Finsler disks with minimizing geodesics

We show that the Holmes--Thompson area of every Finsler disk of radius $r$ whose interior geodesics are length-minimizing is at least $\frac{6}{\pi} r^2$. Furthermore, we construct examples showing that the inequality is sharp and observe that the equality case is attained by a non-rotationally symmetric metric. This contrasts with Berger's conjecture in the Riemannian case, which asserts that the round hemisphere is extremal. To prove our theorem we discretize the Finsler metric using random geodesics. As an auxiliary result, we show that the integral geometry formulas of Blaschke and Santal\'o hold on Finsler manifolds with almost no trapped geodesics.