Schiffer's Conjecture and the Pompeiu Problem

If a smooth bounded domain in $\mathbb{R}^n$ admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via Neumann eigenfunctions is the version on Yau's 1982 list (Problem 80), and Williams proved the two formulations logically equivalent for simply connected domains in 1976. Cao-Labora and de Dios Pont construct infinitely many planar domains with large $N$-fold symmetry that are not balls and admit such an eigenfunction, disproving Schiffer's conjecture; applying Williams' classical reduction to the same domains (their Corollary 1.2) disproves Pompeiu's problem as well.