I would like to suggest the idea that If a capital city falls, then the realm of the attacker will get a percentage of the fallen realm's authority gains for a certain amount of time, be it a week or a month, but make it a real tax so the attacked realm is weaker afterwards. The ones behind the siege gain more from letting the realm rebuild itself and earn them authority (Which I hope i important enough for people to choose to do that)
The one thing I'm not sure about is how to split the taxes among the attacking realms? seeing as recent sieges have been the combined work of more than one faction