Uniqueness and convexity conjecture for optimal densities

Let $\Omega$ and $k$ satisfy the assumptions in Section 1. Suppose also that $\Omega$ is convex and $\rho_->0$. An optimal-density uniqueness and convexity conjecture asserts that the optimal density for the density formulation is unique and that $B_+$ is convex. This conjecture concerns the geometry and uniqueness of maximizers for the maximal energy problem under convexity and positivity assumptions; the supplied text gives no resolution, so the claim remains open.