Conjecture on injectivity of the ground-state beta ratio

For each ground state soliton $P_\omega$ of the cubic-quintic nonlinear Schrödinger equation on $\mathbb{R}^3$, define

$$\beta(\omega)=\beta(P_\omega)=\frac{\int |P_\omega|^6\,dx}{\int |\nabla P_\omega|^2\,dx}.$$

Injectivity conjecture. The mapping $\omega\mapsto\beta(\omega)$ is injective; equivalently, ground state solitons are uniquely identified by the ratio $\beta(\omega)$. The claim is presented as compellingly supported by numerical investigations, but no proof or disproof is supplied in the source.