Constant Width Curves - Packing

A quote by H L Resnikoff: https://arxiv.org/pdf/1504.06733.pdf

I conjecture that the packing by Reuleaux triangles has the greatest density for packings of the plane by a curve of constant width. Moreover, it seems likely that, given two curves of the same constant width, the one with the smaller area has the greater maximal packing density. The reason is that the curve that bounds the smaller area will be ‘less round’ and offer more opportunities to snuggle closer to its neighbors.

The above conjecture seems to imply that the circle has least density of packing among constant width curves. The construction given by Martin Gardner (Colossal Book of Math, page 38) for constant width curves with unequal arcs puts a bit of doubt on this. Indeed, I am not sure if the worst packer of the plane among convex regions is necessarily a curve with central symmetry (the present best candidate with central symmetry is the 'smoothed octagon').