The multi-phase concentric-balls conjecture

Let $m\in\mathbb{N}$ and let $\Omega_k$, $k\in\{1,\dots,m\}$, be as in the introduction. Consider the multi-phase overdetermined problem described by problem

, with outer boundary $\partial\Omega_m$ and normal derivative $\partial_n$. **The concentric-balls conjecture.** Problem

admits a solution $u$ of class $C^m$ in a neighborhood of $\partial\Omega_m$ satisfying

$$(\partial_n)^k u\equiv c_k\quad\text{on }\partial\Omega_m\quad (k=1,2,\dots,m)$$

for some constants $c_k\in\mathbb{R}$ if and only if the sets $\Omega_k$ are concentric balls. The source explicitly describes this conjecture as false: the preceding result establishes the corresponding assertion for $m=2$, but the proposed generalization fails for higher phases.