How to Complete Rumors Abound
You have to defeat all foes to complete this quest. The law forbids copycat, which means that you are not allowed to do the same thing as the unit before you did. It doesn't matter if the previous unit is friend or foe; your unit is not allowed to do what that friend or foe did.
The law does not apply to Moving or Waiting, so you can do either of those any time and not get penalized.
It's easier to avoid breaking this law if you don't have more than one unit with the same job, and if use your units' specialized moves, specific to their jobs. That way, even if you forget about the law, it is less likely that your units will have the same specialized moves in their different jobs.
This battle features some undead enemies: the Ghosts, Zombies, and Wraith. Undead enemies can be damaged by potions and Phoenix Down, as well as by healing magic. When you reduce their HP to 0, they do not disappear from the battlefield. Instead, they turn into a gravestone with stars floating overhead, and they are in the "0 HP" status. After three turns in 0 HP status, the unit will come back to "life". However, you can permanently get rid of undead by using a Phoenix Down on the gravestone, or using moves like Burial (an Archer A-Ability) or Exorcise (a White Monk A-Ability). If you use those moves on a 0 HP unit, they are guaranteed to work. But if the unit is still active, those moves only have a certain percent chance of working.
If all the undead units are gravestones when you defeat the Floating Eyes, the battle will end. Or you can use the methods mentioned in the previous paragraph to permanently get rid of the undead enemies.