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I thought there was Velcro on those flaps. If not then a little contact cement on both surfaces, let it dry then push it in place should work if your rope wedge doesn't.
After some high speed highway travel with the top down today, we have success. The rope wedge appears to be holding and I recommend it as a way in which to remedy the situation.
Your photos were very helpful, but I'm hoping you can advise as to how you re-installed the headrest after removal. I just repaired my driver's side headrest but I'm having problems reinstalling it. It fits back into the holes perfectly, but now it just slides up and down with little to no resistance. I can't get it to recattach to the electric up/down control. Is there some trick to reattaching it to the motorized raise/lower mechanism?