Jan, Here's the link [waaay below] to the RECOGNIZED Mayflower Passenger list. Click through The Edward Fuller line and you will see Matthew Fuller WAS recognized by GSMD nut is not now. They dispute all the arguments made below. Why, they don't say.
Anderson in Mayflower Migration has Matthew as Edward's son [MM 88-89]. Matthew came over later, after the 1623 land division.
See TAG 61-198-99 and Silver Book MF 4:5-6. As to Matthew and his descendants, Anderson recommends as EXCELLENT Paul Prindle, Ancestry of Elizabeth Barrett Gillespie ... (n.p. 1976) 157-162. Answers likely in Prindle's book.
I have heard NOTHING about this GSMD position and nothing has been put in print at AmAnc. in this regards. Perhaps someone else has news.
My $ are on Anderson following Bruce McGonnigle, author of SBMF4.
P.S. Ah! From one of the several John Fuller profiles, I see that this GSMD rejection has to do with DNA analysis. Could not find press release about at GSMD. Onto Wiki which says It could be Matt's Mom is Edward's SECOND wife. Details at Wiki. If THAT'S true [info suggestive only], why, yes, Matt descendants would be OK again.
GSMD Recognized Passengers