The eigenvalue-branch bound for the Dirichlet-to-Neumann map

Let $\Omega\subset\mathbb{R}^d$ be a bounded Lipschitz domain, and let $\sigma^{(\Lambda)}$, $\Lambda\le 0$, be a fixed analytic eigenvalue branch of the Dirichlet-to-Neumann operator ${\mathcal D}_\Lambda$. Eigenvalue-branch bound. One should have

$$\sigma^{(\Lambda)}-\sigma^{(0)}\le \sqrt{-\Lambda}\qquad\text{for all }\Lambda\le 0.$$

This is proposed as a generalisation of the established bound for the first eigenvalue, but the source gives no resolution of the conjecture.