I checked the pics on Google Images, they're virtually taken recently with nothing dating back to the Ottoman period to validate this claim. All look like black Arabian abayas and burkas and it's difficult to imagine that çarşafs and abayas are identical like that.

Fun fact, "çarşaf" or more accurately "şarşaf" in Egyptian Arabic means table clothes or bed clothes. It's amazing words and consequently languages evolve and take on meanings over time.