Layer potentials provide a classical approach to boundary value problems for Laplace’s equation on Lipschitz domains. Kenig’s 1994 spectral-radius conjecture would guarantee operator-norm convergence of the Neumann series for inverting the shifted adjoint double-layer operators $D_\Gamma^{*} \pm\frac{1}{2}I$ on mean-zero $L^2$ densities, when the boundary $\Gamma$ is connected. In this talk, I will present a counterexample: a bounded simply connected planar Lipschitz domain whose double-layer operator on arclength $L^2$ has essential spectral radius strictly greater than $1/2$. The construction combines oscillations at separated scales with a computer-assisted inequality for $2\times2$ Hermitian matrices. We will see how growth under refinement and the insertion of rescaled graph segments produce approximate eigenvectors on a single boundary, while keeping the graph slopes uniformly bounded.