Complexity of flip distance for almost perfect matchings in convex point sets

Determine the computational complexity of the flip distance problem for almost perfect matchings on point sets in convex position, where an almost perfect matching is a non-crossing matching with exactly one unmatched vertex and a flip removes one edge and adds another edge such that the result is again an almost perfect matching.

Background

The paper surveys flip-graph reconfiguration problems beyond triangulations, including matchings. For convex point sets, perfect matchings have polynomial-time computable flip distance, whereas in general position the problem is NP-hard.

For almost perfect matchings (one isolated vertex), NP-completeness is known in general position, but the status in convex position is not settled; the authors explicitly mark the complexity as open.