Complexity of flip distance for non-crossing spanning paths in general position

Determine the computational complexity of computing the flip distance between two non-crossing spanning paths on a planar point set in general position, where a flip removes one path edge and adds another edge so that the graph remains a non-crossing spanning path.

Background

The authors note that for convex point sets, the flip distance between plane spanning paths is solvable in polynomial time.

They explicitly state that the complexity in general position is still open and speculate it may be NP-hard, but no classification is currently known.