I personally do not like white lithium grease when it dries out, it becomes worse than a lack of lubricant and turns into glue. PO of my TD used this on the drivers rear door for the door check, it froze open one day when temps were below freezing. Had to take the damn door apart and spray it with brake clean to get it free.

I would use something like Aerokroil or ATF as a lube, it won't seize anything later. I realize the mechanisms between the two are different, but....