The hypergraph Bunkbed conjecture

Let $G=(V,E)$ be a finite graph, let $T\subseteq V(G)$, and let $\mathcal U=\{U_1,\ldots,U_k\}$ be a partition of $E$ into connected subgraphs. In the model $E_4^{T,\mathcal U}(G)$, all edges in each $U_i$ receive the same color, red or blue, independently and with equal probability; a walk may change color only at a vertex in $T$.

Hypergraph Bunkbed conjecture. For every $u,v\in V(G)$,

$$P(u_0\longleftrightarrow v_0)\ge P(u_0\longleftrightarrow v_1).$$

This generalizes the two-color model by allowing connected groups of edges to share a color, and is needed for the paper's outerplanar-graph argument. The assertion is not established in general.