Closed nodal surfaces for simply connected domains in higher dimensions

We give an example of a domain in dimension $N\ge 3$, homeomorphic to a ball and with analytic boundary, for which the second eigenvalue of the Dirichlet Laplacian has an eigenfunction with a closed nodal surface. The domain is constructed via a sequence of perturbations of the domain of S. Fournais [J. Differential Equations $\mathbf{\text{173}}$ (2001), 145-159].