Do conditional triggered abilities check their condition again before resolving?
Once an ability has been triggered, that's it: it's on the stack and you don't have to worry about the triggering condition anymore.

You can see this in the Gatherer rulings for Grenzo&#39;s Cutthroat itself:


  5/29/2014     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.


The exception to this is the "intervening if" clause, which is specialized rules text in the form "When/Whenever/At [trigger event], if [condition], [effect]."