I really don't understand how or why the Besieged bosses suddenly start taking damage. It seems to possibly be timed to the percentage of enemies left, I'm not sure. Whatever it is, they only seem to start taking damage like a minute before everyone just warps away, so the window for killing them is incredibly small. I don't know if this is intentional or not.
It's also worth mentioning that up until whatever the trigger is that allows them to take damage, they also completely resist all debuffs even dia and inundation.