Hislop–Lutzer exponential decay conjecture for gamma-harmonic liftings

Let $\{\phi_k\}_{k=1}^{\infty}$ be eigenfunctions of the Dirichlet-to-Neumann operator, with corresponding eigenvalues $\lambda_k$, and let $v_{\phi_k}$ denote the $\gamma$-harmonic lifting of $\phi_k$ to $\Omega$. For a compact set $\mathcal{C}$ in the interior of $\Omega$, write $\operatorname{dist}(\mathcal{C},\partial\Omega)$ for its distance to the boundary. Hislop–Lutzer conjecture. In fact the decay is exponential rather than algebraic:

$$|v_{\phi_k}(x)|=O\left(\exp\left[-k\,\operatorname{dist}(\mathcal{C},\partial\Omega)\right]\right).$$

This conjecture strengthens the preceding algebraic decay estimate obtained from the Weyl asymptotic formula. The source attributes the exponential assertion to Hislop and Lutzer; its resolution is not specified here.