No-hot-spots conjecture for simply connected planar mixed-boundary triples

Let $(\Omega,D,N)$ be a triple where $\Omega$ is simply connected, $D$ is a Euclidean line segment, and

$$N=\partial\Omega\setminus\overline{D}.$$

Say that $(\Omega,D,N)$ has no hot spots when each first mixed eigenfunction for the Dirichlet-Neumann problem on $(\Omega,D,N)$ has no interior local extrema. No-hot-spots conjecture. Every such triple $(\Omega,D,N)$ has no hot spots. This conjecture concerns the mixed Dirichlet-Neumann analogue of the hot spots problem; the paper notes that examples with hot spots exist for more general triples, so the stated geometric hypotheses are essential, and the conjecture is motivated by numerical evidence.