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

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