The global minimization conjecture for decay rates in layered resonators

Let $b_1$ and $b_2$ be constants satisfying $0\le b_1<b_2<+\infty$. For each admissible frequency parameter $\alpha$, let $\beta(\alpha)$ denote the minimal decay rate among the corresponding quasi-eigenvalues, and let $\mathbb{A}$ be the admissible class of layered structures with quasi-eigenvalue set $\Sigma[\mathbb{A}]$. Global decay-rate minimization conjecture. The inequality

$$\liminf_{|\alpha|\to\infty}\beta(\alpha)<\beta(\alpha_0)$$

holds for every admissible frequency $\alpha_0$. In particular, the global infimum

$$\inf_{\omega\in\Sigma[\mathbb{A}]}|\operatorname{Im}\omega|$$

is not attained over $\mathbb{A}$. The conjecture is motivated by numerical observations that higher-frequency quasi-eigenvalue clouds become wider and successively approach the real axis; the paper notes that a stronger version assuming existence of $\lim_{|\alpha|\to\infty}\beta(\alpha)$ could explain sliding effects in gradient-ascent computations, but gives no resolution.