Healing/Tank on PS3 or PS4

[BLaCKnBLu3B3RRY]

I'm pretty sure you don't gain aggro based on what you do while a mob is sleeping, regardless of whether it's engaged or not, so that wouldn't change. (The casting of the sleep spell is considered to complete in the instant before the mob falls asleep, so that spell itself gives the BLM or whoever cast it some aggro, but not a healer he's partied with.) The mob should wake up with whatever aggro it had the moment it fell asleep.

the tank runs up to a mob of 3 linked enemies. it doesn't matter if he/she only engaged one target or all of them. the BLM uses Deep Sleep on the mob. if none of the other enemies are attacked they will all remain asleep. in this situation only the enemy currently being attacked by the tank will be awake. now if those enemies were put to sleep BEFORE the healer does anything to the tank. then those enemies will remain in an unengaged state for the healer. they won't even show up in your Enmity List. so this means it doesn't matter how much you heal the tank. you will induce ZERO enmity on the sleeping enemies. again, provided they were put to sleep before the healer engaged.

however, if they are in your Enmity List, other words you used a action on the tank before they were put to sleep, then obviously you induce enmity. regardless if they are sleeping or not.