Uniqueness and convexity conjecture for optimal configurations

Let $\Omega$ be a convex domain, and let $D=D_{\alpha,A}$ be an optimal configuration in $\Omega$. Write $D^c$ for its complement and let $\overline{\alpha}_\Omega(A)$ denote the threshold appearing in the optimization problem. Uniqueness and convexity conjecture. If $\Omega$ is convex then $D$ is unique, and $D^c$ is convex, at least when $\alpha\leq\overline{\alpha}_\Omega(A)$. The authors prove convexity for small $\alpha$, but the full uniqueness and convexity assertion remains open.