I went because they have many years of experience and are said to do a lot of eye surgeries.
They said the existing scar is deep, so rather than double-line excision, it should be excised. When I asked if my eyes lacked extra skin, they pinched it with forceps and said it's not that there isn't any.
It is done under local anesthesia, and there is an added cost if you want sedation anesthesia.