ULC rank sequences in the random-cluster, competing-urns and exclusion models

In the random-cluster (RC) model and the competing urns model, let $X_e$ denote the indicator associated with an edge or site, and let $S$ be any subset of the relevant index set. In the exclusion model, let the occupied-site indicators be observed in a set $S$ at time $t$. ULC model conjecture. In the RC model, the rank sequence of

$$\sum_{e\in S}X_e$$

is ULC for every subset $S$; the same is true in the competing urns model. In the exclusion model, the total number of occupied sites in $S$ at every time $t$ has a ULC rank sequence. These model-specific assertions were presented as conjectures, with the spanning-tree case noted as a subcase of the RC assertion.