  We exhibit a Finsler metric on the 2-sphere whose systolic (Holmes-Thompson)
ratio is $\frac{4{\pi}}{3}$. This is bigger than the conjectured maximal
Riemannian systolic ratio of $2\sqrt{3}$ achieved by the Calabi-Croke metric.
The construction of the Finsler metric is heavily inspired by a paper of
Cossarini-Sabourau.
