Akl et al.'s connectivity conjecture for flip graphs of plane spanning paths

Let $S$ be a point set in general position. A plane spanning path on $S$ is a noncrossing straight-line spanning path, and the flip graph has these paths as vertices, with adjacency defined by a flip. Akl et al.'s conjecture. The flip graph of plane spanning paths on $S$ is connected. Connectivity is known for convex point sets and for point sets with at most two convex layers, but remains open for point sets in general position.