On the Robin eigenvalues of the Laplacian in the exterior of a convex polygon

Let $\Omega\subset\mathbb{R}^2$ be the exterior of a convex polygon whose side lengths are $\ell_1,\dots,\ell_M$. For a real constant $\alpha$, let $H_\alpha^\Omega$ denote the Laplacian in $\Omega$, $u\mapsto -\Delta u$, with the Robin boundary conditions $\partial u/\partial\nu=\alpha u$ at $\partial\Omega$, where $\nu$ is the outer unit normal. We show that, for any fixed $m\in\mathbb{N}$, the $m$th eigenvalue $E_m^\Omega(\alpha)$ of $H_\alpha^\Omega$ behaves as $E_m^\Omega(\alpha)=-\alpha^2+\mu_m^D+\mathcal{O}(\alpha^{-1/2})$ as $\alpha\to+\infty$ where $\mu_m^D$ stands for the $m$th eigenvalue of the operator $D_1\oplus\cdots\oplus D_M$ and $D_n$ denotes the one-dimensional Laplacian $f\mapsto -f''$ on $(0,\ell_n)$ with the Dirichlet boundary conditions.