The arboreal-gas bunkbed conjecture

Let $G$ be a graph, let $\widetilde{G}$ be its associated bunkbed graph, and let $\mu^{\text{F}}_{\widetilde{G},\lambda}$ denote the arboreal gas measure with parameter $\lambda$. For vertices $u,v\in V(G)$, write $u_i\leftrightarrow v_j$ for connectivity in the bunkbed graph. Arboreal-gas bunkbed conjecture. For every graph $G$, every $\lambda$, and every pair of vertices $u,v\in V(G)$,

$$\mu^{\text{F}}_{\widetilde{G},\lambda}(u_1\leftrightarrow v_1)\geq \mu^{\text{F}}_{\widetilde{G},\lambda}(u_1\leftrightarrow v_2).$$

The arboreal gas measure is a forest-valued weak limit of random cluster measures. The analogous random-cluster bunkbed statement is false for some ranges of $q$, whereas this forest version is presented as a conjecture and is proved only in special cases.