Symmetry-preservation conjecture for small penalization

Let $\Omega$ be a domain, let $A$ be the prescribed parameter, and let $D=D_{\alpha,A}$ be an optimal configuration. Say that $D$ has the same symmetries as $\Omega$ when every symmetry of $\Omega$ is also a symmetry of $D$. Small-$\alpha$ symmetry-preservation conjecture. For any domain $\Omega$ and any $A$, there is $\alpha_0(A,\Omega)$ such that, whenever $\alpha\leq\alpha_0(A,\Omega)$, every optimal configuration $D$ has the same symmetries as $\Omega$. The conjecture concerns symmetry preservation in general, including non-convex domains not covered by the preceding theorem, and remains open.