Characterization of strongly Tutte-descriptive activities

Let $G$ be a graph and let $\psi$ be an activity. The activity $\psi$ is strongly Tutte-descriptive if it is Tutte-descriptive and induces a partition

$$2^{E(G)}=\biguplus_{T\text{ spanning tree of }G}[T\backslash \psi(T),T\cup \psi(T)].$$

Characterization of strongly Tutte-descriptive activities. For any graph $G$, an activity $\psi$ is strongly Tutte-descriptive if and only if there exists a decision tree $\Delta$ such that $\psi$ equals the $\Delta$-activity.

The conjecture seeks to characterize precisely the activities arising from decision trees among those inducing the stated partition of the subgraphs. The right-to-left implication has already been proved in the paper through Theorem and Theorem; the converse is therefore the unresolved direction in the source.