Vector-field expansion conjecture for arborescence ratios

Let $\Gamma$ be a graph and let $\tilde{\Gamma}$ be a $k$-fold cover of $\Gamma$. Let $\mathcal V$ be the set of vector fields of $\Gamma$, let $\operatorname{wt}(\gamma)$ denote the weight of a vector field, and let $f$ be a function from $\mathcal V^{k-1}$ to $\mathbb Z_{\geq 0}$. Vector-field expansion conjecture. The arborescence ratio satisfies

$$\frac{A_{\tilde v}(\tilde\Gamma)}{A_{\tilde v}(\Gamma)}=\sum_{(\gamma_1,\ldots,\gamma_{k-1})\in\mathcal V^{k-1}}f(\gamma_1,\ldots,\gamma_{k-1})\prod_{i=1}^{k-1}\operatorname{wt}(\gamma_i).$$

The conjecture proposes a nonnegative expansion in products of $k-1$ weighted vector fields. The source presents this as a potential combinatorial interpretation of the arborescence ratio; no general proof or resolution is supplied, and the denominator is written with $A_{\tilde v}(\Gamma)$ in the source.