Random-cover expectation conjecture for arborescence ratios

Let $\Gamma=(E,V)$ be a graph, fix a vertex $v$ with non-trivial arborescence, and let $\mathcal V$ be the set of vector fields of $\Gamma$. Let $\Gamma'$ be a uniformly random $k$-fold cover of $\Gamma$. Random-cover expectation conjecture. The expected arborescence ratio is

$$\mathbb E\left[\frac{\mathcal A_{v'}(\Gamma')}{\mathcal A_v(\Gamma)}\right]=\frac{1}{k}\left(\sum_{\gamma\in\mathcal V}\operatorname{wt}(\gamma)\right)^{k-1}=\frac{1}{k}\prod_{w\in V}\left(\sum_{\alpha\in E_s(w)}\operatorname{wt}(\alpha)\right)^{k-1}.$$

This conjecture is motivated by the claim that, over all $k$-fold covers, the ratios exhaust all possible $(k-1)$-tuples of vector fields. That exhaustion is known only in the 2-fold case, leaving the general random-cover expectation open.