Kenig's spectral radius conjecture for the double-layer operator

Let $\Omega_-$ be a bounded Lipschitz domain with connected boundary $\Gamma=\partial\Omega_-$. Let $L^2_0(\Gamma):=\{\phi\in L^2(\Gamma):\int_\Gamma\phi\,\mathrm{d}s=0\}$, and let $D'_{\Gamma}$ denote the adjoint double-layer operator. For a bounded operator $T$ on a Banach space $Y$, write $\rho(T;Y)$ for its spectral radius.

Kenig's conjecture. The spectral radius of $D'_{\Gamma}$ on $L^2_0(\Gamma)$ satisfies

$$\rho(D'_{\Gamma};L^2_0(\Gamma))<\frac{1}{2}.$$

This is the 1994 formulation attributed to Kenig and concerns the spectrum of the double-layer operator on mean-zero boundary data. The paper studies this conjecture for locally dilation-invariant Lipschitz boundaries and gives numerical evidence in the examined two-dimensional class; its general status is not resolved in the supplied text.