Simple digraph counterexample for the fully directed bunkbed model (E8)

Demonstrate the existence of a simple directed graph D for which the bunkbed inequality fails under the fully directed model E8: Form the bunkbed digraph by taking two copies of D, add both directed vertical edges v^(0)→v^(1) and v^(1)→v^(0) for every vertex v, then retain a uniformly random subset of all directed edges; prove that for some vertices u and v in D one has Prob(u^(0) → v^(0)) < Prob(u^(0) → v^(1)), i.e., that BBC(E8, D) is false.

Background

The paper proves that several natural generalisations of the bunkbed conjecture are false, including for directed multigraphs under model E7 where parallel edges are allowed and vertical edges are bidirectional. To investigate whether this negative result persists without multiple edges, the author introduces a fully directed model E8 for simple digraphs, where both vertical directions are present and a uniformly random subset of edges is retained.

The author conjectures that the multiedge and undirected-vertical-edge features of the constructed counterexample are not essential and posits that a simple digraph should also violate the bunkbed inequality under E8.