This problem is unsolved. A simple solution called the "Shephard piano" has area 2.2074+, but this can be improved upon with local modifications. A solution exists with area 2.215649+. It is known that a maximum area exists, but not whether the shape achieving it is symmetric, smooth, or even unique.

See Problem G5 in Croft, Falconer, and Guy, *Unsolved Porblems in
Geometry*, Springer-Verlag, 1991.