Nodal-line conjecture for a simple third Dirichlet eigenfunction

Let $\Omega\subseteq\mathbb{R}^2$ be a connected planar domain, and let $u_3$ be an eigenfunction corresponding to a simple third eigenvalue $\lambda_3$ of the Dirichlet Laplacian on $\Omega$. A nodal line is a connected component of the zero set of $u_3$. Nodal-line conjecture. Not all nodal lines of $u_3$ are closed.

The paper calls this conjecture unproved but very plausible and uses it in the argument concerning local maximizers of $\lambda_3/\lambda_1$.