Kasteleyn's Bunkbed Conjecture for Bernoulli bond percolation

Let $G=(V,E)$ be an undirected graph, and let $(V^-,E^-)$ and $(V^+,E^+)$ be two copies of $G$. For each $v\in V$, write $v^-$ and $v^+$ for its copies, and form the bunkbed graph by adding bidirected vertical edges between $v^-$ and $v^+$ for every $v\in V$. Under $p$-Bernoulli bond percolation, let $x\leftrightarrow y$ denote the event that the percolated bunkbed graph contains a path connecting $x$ and $y$. Kasteleyn's Bunkbed Conjecture. For every undirected graph $G=(V,E)$ and every $u,v\in V$,

$$\mathbb{P}_p(u^-\leftrightarrow v^-) \geq \mathbb{P}_p(u^-\leftrightarrow v^+).$$

The conjecture asserts that the two vertices on the same floor should be at least as likely to be connected as vertices on opposite floors. It was disproved by Gladkov, Pak and Zimin, whose counterexample was based on a graph found by Hollom.