Do analytic eigenvalue branches of the Robin problem converge?

Question

Consider the Robin Laplacian eigenvalue problem on a bounded domain $\Omega\subseteq \mathbb{R}^2$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\alpha u=0$ on $\partial\Omega$. It is well known that each ordered eigenvalue $\lambda_n(\alpha)$ converges to the $n$th Dirichlet eigenvalue as $\alpha\to\infty$. However, there is a difference between ordered eigenvalues and analytic eigenvalue branches converging. Does anyone know of a proof (or counterexamples) of the latter?