Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters

Let $\Omega \subset R^n$ with $n⩾2$ be a bounded Lipschitz domain with outer unit normal $\nu$. For $\alpha \in R$, let $R_{\Omega }^{\alpha }$ be the Laplacian in $\Omega$ with the Robin boundary condition ${\partial }_{\nu }u+\alpha u=0$, and denote by $E\left(R_{\Omega }^{\alpha }\right)$ its principal eigenvalue. In 2017, Bucur, Freitas, and Kennedy stated the following open question: Does the limit of the ratio $E\left(R_{\Omega }^{\alpha }\right)/{\alpha }^2$ for $\alpha \to -\infty$ always exist? We give a negative answer.