Grenzo's Cutthroat

Dethrone doesn't trigger if the creature attacks a planeswalker, even if its controller has the most life. The same is true if the creature attacks a battle, even if its protector has the most life.

Once dethrone triggers, it doesn't matter what happens to the players' life totals before the ability resolves. You'll put a +1/+1 counter on the creature even if the defending player doesn't have the most life as the ability resolves.

In a Two-Headed Giant game, dethrone will trigger if the creature attacks either player on the team with the most life or tied for the most life.