The so-called Schiffer conjecture was stated by S.T. Yau in his famous list of open problems as follows: If a nonconstant Neumann eigenfunction $u$ of the Laplacian on a smooth bounded domain in $\mathbb{R}^2$ is constant on the boundary, then the domain is a disk. In this talk we will consider a version of such question for domains with disconnected boundary. Specifically, we consider Neumann eigenfunctions that are locally constant on the boundary, and we wonder if the domain has to be necessarily a disk or an annulus. We will show that the answer to the above question is negative. Indeed, there are nonradial Neumann eigenfunctions which are locally constant on the boundary of the domain. The proof uses a local bifurcation argument together with a reformulation of the problem by Fall, Minlend and Weth that avoids a problem of loss of derivatives. This is joint work with A. Enciso, A. J. Fernández and P. Sicbaldi.