Payne's nodal line conjecture for planar domains

Let $\Omega\subset\mathbb R^2$ be a bounded domain, and consider the second eigenfunction of the Laplacian on $\Omega$ with Dirichlet boundary condition. Payne's nodal line conjecture. The second eigenfunction does not have a closed nodal line. The conjecture concerns the topology of the nodal set and the energy-minimising two-partition formed by the second nodal domains; its status is not resolved in the supplied source.