Super-exponential decay conjecture for the orthogonality parameter

Let $\sigma_T(\kappa,H,ell)>0$ be the parameter satisfying

$$\|g\|_{Y^\prime}\coloneqq \sup_{v \in Y} \frac{{( g\,,\,v )}_{L^2(\omega)}}{{\| v \|}_{\mathcal{V}_\omega}} \leq \sigma_T(\kappa,H,ell)\|g\|_{\mathcal V_{\omega}^\prime}.$$

Here $\sigma_T$ measures the quasi-orthogonality of $g$ on $Y$, and depends on the wavenumber through the wavenumber-dependent space $Y$ and the norm of the solution space. Super-exponential decay conjecture. The quantity $\sigma_T$ decays super-exponentially in $ell$: there exist constants $C_\mathrm{sd}(\kappa,H,ell)>0$, depending polynomially on $\kappa$, $H$, and $ell$ but independent of $T$, and $C>0$, independent of $\kappa$, $H$, $ell$, and $T$, such that

$$\sigma_T(\kappa,H,ell) \leq C_\mathrm{sd}(\kappa,H,ell) \exp\left(-Cell^{\frac{d}{d-1}}\right).$$

This conjecture predicts the super-exponential localization underlying the proposed decomposition and is supported in the paper by a numerical experiment; no proof or resolution is supplied here.