Symmetry-breaking threshold conjecture for annuli

Consider an annulus of thickness $\tau$, and write $\delta=A/|\Omega|$ for the prescribed area fraction. Let $D=D_{\alpha,A}$ be an optimal configuration, and let symmetry breaking refer to failure of rotational symmetry of the optimal configuration. Annular symmetry conjecture. For each $\alpha,\delta>0$, there is $\tau_0(\alpha,\delta)$ such that symmetry breaking occurs for the annulus of thickness $\tau$ if and only if $\tau<\tau_0(\alpha,\delta)$. One direction is proved in the paper, while the converse—symmetry preservation for sufficiently thick annuli—remains open.