Would fat transfer to lips on an Ehlers-Danlos hypermobility patient run the risk of saggy lips?

I have EDS hypermobility but would like a more permanent fuller lip result than injections offer. (I'm not interested in implants.) So I've been considering fat transfer to lips since I had a good experience with BBL. But would my more elastic skin cause fuller lips to sag, or are lips anchored fairly well to the face? I found a surgeon whose results I like but as with many surgeons their experience w/EDS is limited/nonexistent and so he understandably didn't feel comfortable offering an answer.