I did the same 03/04 top/frame replacement a couple of years ago. I can't say for sure, but I think this is a cord that runs from one side to the other and should be tight when the top is closed. It runs through a pocket/sleeve attached to the outer layer of the top and keeps it tight by pulling slightly down when the top is closed.
When I put my new top on, there was some fabric vibration/noise at speeds over 65 or so and I noticed this cord was loose. I re-tied one end to shorten it slightly and the vibration stopped.
If this is the same cord and it comes loose, you may need to shorten it slightly. If I remember right, on my car the cord went through a hole in the frame, then had a knot to set the length.
I hope that is helpful.
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