If the ballast outputs feature DC blocking capacitor for each lamp (that was not the case just with early ballast designs), then they can not interfere with each other - even when one lamp rectifies, it does not cause any DC current, just DC voltage component across the faulty lamp. And no DC current means no electrolysis effects (mercury migration,...) could happen.
But indeed, with new lamps it could be either contamination, or nor properly fired amalgam capsule on one side. But that will get sorted after some hour by itself.
But even the emergency thing would "get sorted" after some time by itself: The lamp will just shut down completely (or the emergency units will switch it back onto the normal ballast due to the flat batteries - depend, how that is designed)
Does it look like
this?