So while driving with the top up I noticed a piece of material flapping near the front of my rear quarter window. Its about 1-1/2 inch wide and 4 inches long. It looks like a liner that runs along the edge of the top which has delaminated from another piece where the sponge padding has deteriorated. Has anyone dealt with this issue?

 

I had the same problem and the headliner, due to age had deteriorated just as you described. I had a trim shop replace the headliner with an aftermarket kit from CA. It's a tough job and not a job for an inexperienced installer. Total cost $800.