Symmetry-breaking threshold conjecture for dumbbell domains

Let $\Omega_h$ be a dumbbell domain, and let $D=D_{\alpha,A}$ be an optimal configuration. Dumbbell symmetry conjecture. For every $\alpha>0$, there is $\rho_1(\alpha,h)>0$ such that symmetry breaking occurs if and only if $|\Omega|-A<\rho_1(\alpha,h)$. The conjecture describes a sharp transition between symmetry breaking and symmetry preservation in dumbbell domains; the text presents it as motivated by numerical experiments and leaves it open.