Maximal-area reduced bodies in the hyperbolic plane

Determine whether there exists, for a fixed minimal width (thickness) w, any reduced convex body in the hyperbolic plane H^2 whose area exceeds that of the quarter of a disk of radius w.

Background

The paper proves that among ordinary reduced n-gons of given thickness, the regular n-gon maximizes area, and also that all ordinary reduced polygons have smaller area than a circle of the same thickness.

It further observes that the quarter of a disk (also reduced) has area greater than the circle of the same thickness, and then notes the absence of knowledge of any reduced body with larger area, pointing to an unresolved global maximization question among all reduced bodies of fixed thickness in H^2.