Fundamental gap conjecture for stationary states of the dissipative rotational NLS equation

Let $m>0$, and let $\mathcal{S}_m$ denote the set of stationary solutions with total mass $m$. Let $Q_{gs}$ be a ground state, and let $E[Q]$ denote the total energy of a stationary state $Q$. Fundamental gap conjecture. There exists $\delta>0$ such that for any $Q\in\mathcal{S}_m$, either

$$Q=e^{\mathrm{i}\mkern1mu\phi}Q_{gs}$$

for some $\phi\in\mathbb R$, or

$$E[Q]\geq E[Q_{gs}]+\delta.$$

The conjecture asserts a uniform positive energy gap between the ground-state orbit under phase rotations and all other stationary states. Such a gap would support convergence of solutions with sufficiently low initial energy to the ground state, but the source does not establish whether it holds.