The bunkbed conjecture for independent bond percolation

The bunkbed graph $G=(V,E)$ of a graph $\widetilde{G}=(\widetilde{V},\widetilde{E})$ is formed by superposing two identical copies of $\widetilde{G}$ and adding edges between corresponding vertices. Write $G=\widetilde{G}\times\{0,1\}$. A vertex $(x,0)$ is in the bottom graph, a vertex $(x,1)$ is in the top graph, and the symmetrical vertex of $u=(x,i)$ is $u'=(x,1-i)$. The bunkbed conjecture. Let $u$ and $v$ be vertices of the bottom graph, and let $v'$ be the symmetrical vertex of $v$. For every $p\in[0,1]$, independent bond percolation on $G$ satisfies

$$\mathbb{P}_{p}(u\leftrightarrow v)\geqslant\mathbb{P}_{p}(u\leftrightarrow v').$$

The conjecture compares the probability of connecting two vertices on the same level with the probability of connecting corresponding vertices on opposite levels. It is known for some subclasses, including outerplanar and wheel graphs; for the complete graph, the paper proves the claim for $p\geqslant 1/2$, while the general statement remains open.