Small-energy stability in the limiting convex case

Let $m,n$ be the mode indices, let $P$ be the parameter, and let $\Theta_m$ and $\Theta_n$ denote the corresponding modes with energy $E_{\Theta_m}$. Assume

$$0<P=m^2<n^2.$$

Let $E_2$ be the threshold from the theorem describing large-energy stability and instability in this limiting case. Small-energy stability conjecture. There exists $0<E_1\leqslant E_2$ such that, whenever $0<E_{\Theta_m}<E_1$, the mode $\Theta_m$ is linearly stable with respect to $\Theta_n$.

The cited theorem leaves small-energy stability unresolved; this conjecture extends the established large-energy analysis to a nontrivial low-energy interval.