Baring some circulation due to minor friction of the air against the walls the air would remain motionless relative to the cabin. However, since the helicopter has inertia, which according to Newton means it remains at rest until an external force acts on it, the acceleration the cables impart on the elevator will have no immediate effect on the helicopter and it will initially remain motionless and seem to move downward relative to the cabin. As the elevator accelerates the relative speed between the air in the cabin and the helicopter's blades will increase and at some point create enough friction do impart acceleration on the helicopter. However, I doubt this would keep it from hitting the floor first.
https://www.youtube.com/watch?v=NrhFhUUU4SE