Rigidity in the planar Ulam floating body problem with perimetral densities $\sigma=\tfrac18,\tfrac38$ 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 $\sigma=\tfrac18$ or $\sigma=\tfrac38$ is a disk.