On a homothety conjecture for convex bodies of flotation: counterexample

Let $K$ be a convex body in ${\mathbb R^2 }$. For every $\theta\in {\mathbb R}$ and the corresponding unit vector $e(\theta)=(\cos\theta,\sin\theta)$ and for every $t\in {\mathbb R}$, define the half-planes
$$ W^+(\theta,t)=\{x:\,\langle x, e(\theta)\rangle\ge t\}\quad\textrm{and}\quad W^-(\theta,t)=\{x:\,\langle x, e(\theta)\rangle\le t\}. $$
If $0<{\mathcal D}<1/2$, then for every $\theta\in{\mathbb R}$ , there is a unique $t(\theta)$ such that
$$ \textrm{vol}_2(W^+(\theta, t(\theta))\cap K)={\mathcal D}\,\textrm{vol}_2(K). $$
The corresponding convex body of flotation $K^{\mathcal D}$ is defined as
$$ K^{\mathcal D}=\bigcap\limits_{ \theta\in {\mathbb R} }W^-(\theta,t(\theta)). $$

We investigate the homothety conjecture for convex bodies of flotation of planar domains. We show that there is a density close to $\frac{1}{2}$ for which there is a body $K$ different from an ellipse with the property that $K^{{\mathcal D}}$ is homothetic to $K$.