Souam's conjecture on Schiffer domains in the 2-sphere

Let $\Omega\subset S^2$ be a sufficiently regular Schiffer domain, meaning a smooth bounded domain for which the overdetermined Neumann problem $(\textrm{N}_{\mu})$ has a solution for some $\mu>0$. A geodesic disk is a disk bounded by a geodesic circle, and a round symmetric annulus is an annulus bounded by two concentric geodesic circles.

Souam's conjecture. If $\Omega\subset S^2$ is a sufficiently regular Schiffer domain, then $\Omega$ is either a geodesic disk or a round symmetric annulus.

Souam's conjecture follows classification results for certain cases, including simply connected domains with $\mu=2$ and domains with $\mu$ equal to the second Dirichlet eigenvalue. The paper's abstract states that this conjecture is disproved by constructing a related family of subdomains of $S^2$.