I have been looking into grease recently and although any will, do they are all mostly based on oil with extra stuff to make it thicker - or stickyer or better for constant turning (CV ) or better crush properties or better for waterproofing, rubber grease based on veg oil and not petroleum oil and so on.
So I would suggest that the very best you can use would be something better suited to the application which is crushing rather than constant velocity (CV).
Thats just what I have picked up from (too munch ) reading on the subject.
Cause thats what I do ------ I over think things