Background: Among convex plane $n$-gons of unit diameter, we can try to achieve:

• the largest area. (This is called the biggest little polygon with $n$ sides; for $n$ odd, the regular polygon on $n$ sides is known to be the biggest little $n$-gon; for $n=6$ the regular polygon is suboptimal and Ron Graham found an optimal polygon; for $n=8$ the regular polygon is also suboptimal; for higher even $n$ I am not sure).
• the largest width or the largest perimeter. (The Reinhardt polygons maximize both of these.)
• the largest moment of inertia about an axis through the center of mass and normal to the plane (my question here).

Question: Among convex $n$-gons of diameter 1, which maximize this moment of inertia?