Shortest-path bisaturation conjecture for transfer system lattices

Let $G$ be a group, and consider the Hasse diagram of its lattice of $G$-transfer systems. A path has length equal to its number of edges; a shortest path between two transfer systems is a path with the minimal possible length. A transfer system is bisaturated when it has both saturation and cosaturation.

Shortest-path bisaturation conjecture. A shortest path between the trivial $G$-transfer system and the complete $G$-transfer system on a Hasse diagram will pass through as many bisaturated transfer systems as possible.

The conjecture concerns how saturation properties are distributed along shortest paths in transfer system lattices. The examples show that shortest paths need not be unique and need not pass through all bisaturated transfer systems; the conjecture remains open.