The conjecture that non-disc analytic domains have tunneling Steklov problems

Let $\Omega\subset \mathbb{R}^2$ be a bounded, simply-connected domain with real analytic boundary that is not equal to $B(x,r)$ for any $x\in \mathbb{R}^2$, $r>0$. A tunneling Steklov conjecture. The Steklov problem on $\Omega$ is tunneling. The conjecture is motivated by numerical observations for elliptical and kite-shaped domains and by the theorem establishing tunneling for the broader class of BBLCN domains; whether every analytic non-disc domain has a tunneling Steklov problem remains open.