There have been several cures claimed - you will see them if you read through this forum. You need to do a search and really utilize the wealth of material that is here. I mentioned several of the claimed cures in a post, in which I copied and pasted their weblinks, I wrote a couple of months ago. One of the claimed cures was from L-Lysine. Another claimed cure was from washing with dishwashing soap.
A cure doesn't have to work for everybody to be a cure. Cheilitis is from many, many different causes, so there will be many types of cures.
Finding out what you are having an allergic reaction to, and avoiding it, is also a cure. There was a guy a few weeks ago who posted that he found out he was allergic to mouthwash. A woman last week posted that she was allergic to tea tree oil. In December, I found out 2 things that I am having an allergic reaction on my lips to, and there is now at least 1 vegetable that I've begun to get cheilitis from. When I avoid them, my lips are cured. Voila.
I am sure that many people who leave this forum and never return actually have had their lips get better, but don't take the time to update people on the forum to let them know what happened.