Conjecture on the asymptotic balance of the optimal continuation function

Let $\Gamma\Subset\mathbb{H}_+$ be a smooth, bounded, simple curve, let $z\in\mathbb{H}_+\setminus\Gamma$, and let $u_{\epsilon,z}$ solve

$$({\mathcal K}u)(\zeta)+\epsilon^2u(\zeta)=p_z(\zeta),\qquad \zeta\in\Gamma,$$

where

$$({\mathcal K}u)(\zeta)=\frac{1}{2\pi}\int_\Gamma\frac{i u(\tau)}{\zeta-\overline{\tau}}|\mathrm{d}\tau|,\qquad p_z(\zeta)=\frac{i}{\zeta-\overline z}.$$

Asymptotic balance conjecture. As $\epsilon\to0$,

$$\|u_{\epsilon,z}\|_{L^2(\Gamma)}\simeq\epsilon\|u_{\epsilon,z}\|_{H^2}.$$

This would show that the two normalizations defining the asymptotically optimal error maximizer have the same asymptotic size, so the maximizer can be expressed using either the $H^2$ norm or the $L^2(\Gamma)$ norm.