Maximal systolic ratio conjecture for genus two abelian differentials

Let $(X,d_{\omega})$ be a genus two Riemann surface equipped with the flat metric induced by a holomorphic $1$-form, and let the systolic ratio be the square of the systolic length divided by the area. Let $S$ be the surface described in Figure, whose systolic ratio is

$$\frac{2 \cdot \left(\sqrt{13} -3 \right)^2}{\sqrt{3} \cdot \left(1-\frac{3}{4}(\sqrt{13}-3)^2\right)}.$$

Maximal systolic ratio conjecture. The supremum of the systolic ratio over genus two surfaces $(X,d_{\omega})$ equals

$$\frac{2 \cdot \left(\sqrt{13} -3 \right)^2}{\sqrt{3} \cdot \left(1-\frac{3}{4}(\sqrt{13}-3)^2\right)},$$

and, up to homothety, $S$ is the unique surface that achieves this systolic ratio.

This conjecture identifies the optimal flat geometry among genus two Riemann surfaces arising from abelian differentials, strengthening the comparison with the ten-systole example whose systolic ratio is $1/\sqrt{3}$. Its resolution is not specified in the supplied text.