The Schiffer conjecture for bounded Lipschitz domains

Let $\operatorname{\Omega_S}$ be a bounded Lipschitz domain. Consider the overdetermined scalar Neumann eigenvalue problem

$$-\Delta u=\mu_Su\quad\text{in }\operatorname{\Omega_S},\qquad \partial_nu=0\quad\text{on }\partial\operatorname{\Omega_S},\qquad u=c\quad\text{on }\partial\operatorname{\Omega_S}.$$

Schiffer conjecture. If this problem admits a non-trivial solution, then $\operatorname{\Omega_S}$ is a ball. The conjecture is a well-known open problem in geometric analysis and, through the paper's characterization, is equivalent to the assertion that balls are the only bad domains for the fluid-elastic semigroup.