Rigidity in the planar Ulam floating body problem with perimetral densities $\sigma =\frac{1}{8},\frac{3}{8}$ under central symmetry

We prove that the only planar, centrally symmetric, strictly convex body $K\subset {\mathbb{R}}^2$ with $C^1$ boundary that floats in equilibrium in every orientation for the perimetral densities $\sigma =\frac{1}{8}$ or $\sigma =\frac{3}{8}$ is a disk.