Couple of questions:
1)If mummy paralysis is broken (say by interrupting line of sight) during an encounter and then re-established, should everyone make new saves or should I use the original results?
2)If someone saves initially against the paralysis, are they thereafter immune to it during the encounter?
I am not an Autarch, but:
1) I would allow new saves if more than a few rounds has passed.
2) I would require new saves if more than a few rounds has passed.
As a final thought, I would adjust my definition of "a few rounds" based upon the degree to which that definition seemed to be getting gamed by the players (or by myself, if I was tempted to do so by a round in which everyone made their saves, for example).
As a corollary, mummies generally appear in dungeons in groups of 1d4. If a group of four mummies is encountered, does every person viewing them have to save separately against each mummy, or may they roll just once against the entire group? If they must save against each mummy separately, what then happens when a single mummy breaks line of sight, then reappears in a new location while the rest of the mummies maintain visual contact?