The mass monotonicity conjecture for cubic-quintic solitary waves

In dimension $d=3$, let $\phi_\omega$ denote the ground state solution of the cubic-quintic Schrödinger equation at frequency $\omega$, with $0<\omega<\tfrac{3}{16}$, and let $M(\phi_\omega)$ be its mass. Mass monotonicity conjecture. There exists $0<\omega_*<\tfrac{3}{16}$ such that the map $\omega\mapsto M(\phi_\omega)$ is strictly decreasing for $\omega<\omega_*$ and strictly increasing for $\omega>\omega_*$. This conjecture would imply that the two stability transition frequencies coincide, and it is supported by numerical evidence; its status is not resolved in the supplied source.