Holomorphy of the Dirichlet-to-Neumann semigroup on continuous boundary functions for Lipschitz domains

Let $\Omega\subset\R^d$, $d\geq2$, be a bounded connected Lipschitz domain, and let $C$ be a real symmetric uniformly elliptic matrix with Lipschitz entries on $\overline\Omega$. We prove that the weak Dirichlet-to-Neumann semigroup on $C(\partial\Omega)$ is holomorphic of angle $\pi/2$ and bounded on every strictly smaller sector. This answers the continuous-function question in the real symmetric case of Question 4 of ter Elst and Ouhabaz. The proof obtains a supremum norm resolvent bound by boundary rescaling. A complex Robin uniqueness lemma on a half-space excludes the limiting solution; its proof uses vertical $L^2$ Dirichlet estimates and an iteration of boundary mass bounds.