Existence of an edge inactive in every spanning tree for strongly Tutte-descriptive activities

Let $G$ be a graph with a standard edge, meaning an edge that is neither a loop nor an isthmus, and let $\psi$ be a strongly Tutte-descriptive activity. A spanning tree $T$ has an edge active when that edge belongs to $\psi(T)$.

Inactivity conjecture. There exists an edge $e$ of $G$ such that, for every spanning tree $T$, $e$ does not belong to $\psi(T)$; equivalently, $e$ is active in no spanning tree.

The paper states that this is equivalent to the characterization conjecture above and asks for a proof or counterexample. No resolution is supplied for this formulation.