The nodal-line conjecture for the second Cauchy-process eigenfunction

Let $D\subset\mathbb{R}^2$ be a convex domain symmetric with respect to both coordinate axes. Let $\lambda_n$ and $\varphi_n$ be the eigenvalues and eigenfunctions of the Cauchy process killed upon exiting $D$. A nodal line is the zero set of an eigenfunction in $D$. Nodal-line conjecture. There exists an eigenfunction corresponding to $\lambda_2$ whose nodal line lies on one of the coordinate axes. The source explains that, unlike the Brownian case, the location of the nodal line is not known even for a disk or rectangle, which motivates this conjecture.