  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.
