The famous mask popularly known as the 'mask of Agamemnon' is NM 624. In 1876 a reporter for the newspaper Argolis stated that NM 624 had no moustache. This, along with a view that moustaches in Mycenaean art don't normally have the ends point up, was used by David Traill and William Calder to argue that the mask was a forgery, or at the very least that Schliemann altered it to add the moustache.

Schliemann was a fraud in very many ways, but probably not in this. First, in Schliemann's book published in 1878, the mask clearly has a moustache, so if it was altered it must have happened pretty damn fast. Second, it's been pretty well established now that the mask that Schliemann thought of as the 'mask of Agamemnon' wasn't NM 624 at all, but rather NM 623, which isn't nearly as handsome as 624, but is associated with richer burial goods and definitely doesn't have a moustache.

Here's a handy online piece, and here's another, summing up one of the key bits of research on this, a 2005 article by Oliver Dickinson. Here's another article from 2006, by Giampaolo Graziadio and Elisabetta Pezzi, which is even more handy in that the original article is open access.