The alternate arboreal-gas bunkbed conjecture

Let $G$ be a graph, let $T\subset V(G)$ be a set of posts, and let $\nu^{\text{F}}_{G,T}$ denote the alternate arboreal gas measure on red-blue edge-colourings, restricted so that no cycle is admissible. An admissible path may change colour only at posts. Write $u\leftrightarrow_{RR}v$ (respectively, $u\leftrightarrow_{RB}v$) when there is an admissible path from $u$ to $v$ that starts with a red edge and ends with a red (respectively, blue) edge. Alternate arboreal-gas bunkbed conjecture. For every such $G,T$ and every pair of vertices $u,v$,

$$\nu^{\text{F}}_{G,T}(u\leftrightarrow_{RR}v)\geq \nu^{\text{F}}_{G,T}(u\leftrightarrow_{RB}v).$$

This is the arboreal-gas analogue of the alternate bunkbed inequality. The corresponding percolation conjecture is already false in general, and the source gives an outerplanar counterexample to this arboreal-gas version.