Not graphite? Which specific type/class of grease (I imagine you would want something extremely slick)?

I used some kind of white ceramic grease in a stick I would describe as 75-80% condition; it has been about 10 years with little deterioration since. I've read Lithium grease is better.

I've also read somewhere, though I could never find conclusive information, that Nintendo actually intended for these sticks to have some sort of grease included but was unable to for some reason that wasn't as trivial to explain as "Greedy company wants players to keep buying gamepads"..

I can't find any solid reference, but a small "assemble yourself" toy my daughter received for Christmas recommended applying grease to some cogs during assembly and mentioned that the grease itself could not be included in the product due to varying Customs/Import restrictions across the world.

Usually I see Lithium grease recommended