On Garen: Minion Damage (& turret damage if it doesnt) SHOULD stop Perseverence Passive
Personally I don't see any reason why it shouldn't. He's pretty tanky, he has a silence and some other decent damage abilities as well as an execution. When I trade with a Garen Top lane who takes a cloth armor, 5 hp pots and an ignite, It's quite difficult to out trade him if he's remotely witty. Simply because if he gets you low enough he can ignore the value of your creeps, potentially zone you and continue to have this massive passive heal. You're basically losing out on trades the majority of the time if you can't all in him before 6, because even if you win that particular trade, he can merely regen back passively.
I think it's a great thing that his passive works excellent late game I truly do especially for tanks now a days, however it's a bit too strong early game in the laning phase. A point in time where 1v1 trades matter quite a lot and even the slightest things such as creep damage can mean life or death.