Orbital stability of cubic-quintic ground states in two dimensions

For $d=2$, let $Q_\omega$ denote a cubic-quintic ground state solution at frequency $\omega\in(0,\tfrac{3}{16})$. Orbital stability conjecture. All cubic-quintic ground state solutions are orbitally stable. The conjecture is motivated by results showing that $\omega\mapsto M(Q_\omega)$ is increasing near the endpoint frequencies, together with numerical evidence suggesting monotonicity throughout the interval; the general claim is not established in the supplied text.