Energetic stability threshold conjecture for even and odd periodic waves

For a fixed spatial period $L>0$, let $\tilde{\mathcal{E}}_L$ denote the energy of the periodic waves, and let $\omega_L$ and $\Omega_L$ be the lower endpoints of the parameter intervals for the even and odd waves satisfying the equations referred to as (even-wave) and (odd-wave), respectively. Energetic stability is determined by the sign criterion in Theorem (th-stability).

Energetic stability threshold conjecture. There is $\omega_* \in (\omega_L,1)$ and $\Omega_* \in (0,1)$ such that the even wave satisfying (even-wave) is energetically stable for $\omega \in (\omega_L,\omega_*)$ and unstable for $\omega \in (\omega_*,1)$, whereas the odd wave satisfying (odd-wave) is energetically stable for $\omega \in (\Omega_L,\Omega_*)$ and unstable for $\omega \in (\Omega_*,1)$.

This conjecture formalizes the numerical observation that both even and odd periodic waves are energetically stable for smaller values of $\omega$ and become energetically unstable as $\omega$ approaches $1$. The claim is based on the numerical approximations and the sharp stability criterion stated in the cited theorem; its resolution is not established in the supplied text.