The middle-level graph conjecture for vertex-transitive partial cubes of girth six

Let $G$ be a vertex-transitive partial cube, and suppose that its girth is six.

Middle-level graph conjecture. The graph $G$ is a middle-level graph.

The conjecture proposes that the known middle-level graphs exhaust the vertex-transitive partial cubes of girth six. The classification in the paper establishes the claim for the cases covered by its results, while the broader classification problem for higher-valency vertex-transitive partial cubes remains open.