Is this 10 or 25man and what healers are you using?
Just saw on wowprogress you're 10man, but what healers are you using?
Edit: Looking at your log it seems like your healers don't understand the fight tbh, your best try at 4:34 your paladins were on 55 and 66 stacks.
This is how it should go:
Shard 1: Spriest solo soak - Both paladins spam heal to gain stacks.
Shard 2: (If you go feral dps, you can solo soak this one) Have paladin 1 solo heal shard, paladin 2 helps shaman heal tank.
Shard 3: Paladin 1 solo heal again, by the end of this shard he should have 150-200 stacks, allowing him to easily keep the tank alive on his own for the rest of the fight.
Shard 4: Paladin 2 and Shaman healing shards from now on.
If paladin 1 gets tormented, have paladin 2 help with tank healing and the shaman can solo heal shards while paladin 2 is busy.
The key on heroic is getting 1 healer high stacks early so they can deal with tank healing. First shard both paladins should be on at least 60-70 which is how many your healers are getting over 4:34 atm. Also, solo tank it, it's much easier for healers to focus on one tank if they do the stacks right.