Super-exponential decay conjecture for the localized normal derivative

Let $cvarphi^{\rm loc}$ be the local function on the patch $comega$, let $cgamma_{\partial_n}$ denote the normal-derivative trace on $cpartial\omega$, and let $cell$ be the localization parameter. The quantities $cvarepsilon$, $H$, $T$, and $d$ denote, respectively, the diffusion parameter, coarse-mesh scale, patch-related parameter, and spatial dimension. Super-exponential decay conjecture. The quantity

$$\left\|\gamma_{\partial_n} \varphi^{\rm loc}\right\|_{L^2(\partial \omega)}$$

decays super-exponentially in $cell$: there exist constants $C_{sd}(\varepsilon,H,\ell)>0$ depending on $cvarepsilon$, $H$, and $cell$, but independent of $T$, and $C>0$ independent of $cvarepsilon$, $H$, $cell$, and $T$, such that

$$\left\|\gamma_{\partial_n} \varphi^{\rm loc}\right\|_{L^2(\partial \omega)} \leq C_{sd}(\varepsilon,H,\ell)\exp\left(-C\ell^{\frac{d}{d-1}}\right).$$

The claim is motivated by the numerical experiment described in the source, but the supplied text gives no proof or resolution, so its status remains open.