Wagner's sum-of-squares conjecture for spanning-forest Rayleigh differences

Let $\mathcal G$ be a graph, and let $e$ and $f$ be distinct edges. Write $G=F(\mathcal G;\mathbf y)$ for the generating polynomial of its spanning forests, and let $\mathscr S$ be the collection of sets $S\subseteq E-ef$ such that $S\cup ef$ is contained in a cycle of $\mathcal G$. For each $S\in\mathscr S$, let $\mathscr A(S)$ be the collection of spanning forests $A$ such that $A\subseteq E-ef$ and $S\cup ef\subseteq C\subseteq A\cup ef$ for a unique cycle $C$. Here $\mathbf y^X=\prod_{x\in X}y_x$ and $\Delta G\{e,f\}=G_eG_f-GG_{ef}$. Wagner's sum-of-squares conjecture. For some choice of signs $c(S,C)=\pm1$,

$$\Delta G\{e,f\}=\sum_{S\in\mathscr S}\mathbf y^S\left(\sum_{A\in\mathscr A(S)}c(S,C)\mathbf y^{A-S}\right)^2.$$

This identity would give a sum-of-squares certificate for the nonnegativity of the spanning-forest Rayleigh difference and hence imply that graphs are $\mathbf I$-Rayleigh. The statement is presented as a conjecture in the source; no resolution is supplied here.