The implication from Delta-SOS to Phi-SOS for graphs

Let $\mathcal G$ be a graph. Say that $\mathcal G$ is $\Delta$-SOS if every relevant Rayleigh difference $\Delta G\{e,f\}$ has the required sum-of-squares representation, and say that it is $\Phi$-SOS if the corresponding $\Phi$-polynomials have such representations. Delta-SOS implies Phi-SOS. If $\mathcal G$ is $\Delta$-SOS, then $\mathcal G$ is $\Phi$-SOS. This implication is proposed to bootstrap the sum-of-squares argument through two-sums of graphs; the source gives no proof or resolution.