Witness-tree probability conjecture for fully orderable witness trees

A witness tree is built from bad-event labels, with children of a node labeled $B$ required to have distinct labels $B_1,\dots,B_s$ whose set is orderable to $B$. For a tree-structure $\tau$, let P(\text{\tau appears}) denote the probability that it appears, and let $w(\tau)$ be the corresponding witness-tree weight. Witness-tree lemma conjecture. If this orderability condition is enforced at every node of $\tau$, then

$$P(\text{$\tau$ appears})\leq w(\tau).$$

The paper explains that this would strengthen the analysis of the Swapping Algorithm by controlling witness trees at every node rather than only at the root. Establishing the bound requires fine control over the temporal order of resamplings, which the authors were unable to maintain; the conjecture is presented as an open extension of their witness-tree lemma.