Hi
I looked at my car & see that the opening & closing of that door is all mechanical. A spring opens it & a cable attached to the rear lid closes it when the top is open. If it isn't closing then I would think that the cable has come loose from it's mounting positions. I couldn't find a picture of it in the 2010 FSM but I did find one in the 2011 manual. It should be the same. Take a look here
Factory Service Manuals - NICO Club Go to pages 65 - 67
I would think that a good starting point would be to partially open the top & stop it before the working door closes. Then you can compare the two & see if you can see the problem.
Good luck
Dan