I am pretty sure this is impossible. I have been tweaking for a day and the rectangle can't slide back to the goal area.
I'm thinking it's impossible too. You can't roll it in on wheels because the gap isn't high enough. Next option is to have the goal piece spinning toward the goal, but again I don't think there's enough room. You could push the goal piece with wheels, but it's quite heavy and would need LOTS of wheels to push it. I guess there's a tiny, tiny chance you could push it in with the static rectangle , if you could get that rolling on wheels...
Hmm, theoretically, here's a proof of concept: http://fantasticcontraption.com/?designId=9946365
This is my best true effort at recreating that: http://fantasticcontraption.com/?designId=9946490