No-interior-extrema conjecture for first mixed eigenfunctions

Let $\Omega$ be a simply connected Lipschitz domain, and let $D$ be a line segment in the boundary. Consider the first mixed Dirichlet-Neumann eigenvalue problem, with Dirichlet condition on $D$ and Neumann condition on the remaining boundary. A first mixed eigenfunction is an eigenfunction corresponding to the first mixed eigenvalue. No-interior-extrema conjecture. Each first mixed eigenfunction has no interior extrema. This conjecture concerns the location of extrema for first mixed eigenfunctions and is presented as an open problem for general simply connected Lipschitz domains.