This might be a sticky (hope it will be).

Service Manual calls for "Lithium Soap Glycol Grease" for lubing up the sliding pins on the calipers. No dealer uses this, said they can't find it but will use silicone instead. I've found that the silicone will "gum up" and not be lubricating anymore. (caused my pins to stick and a sliding pin to be burnt).

I searched and found it from Toyota (about $18 a tube), which I already used on a brake job I did and now the brakes feel excellent!

Here is the Toyota part number, 08887-01206 Rubber Grease

Rubber grease is how the dealer will come up with a description for it. I got a tube of this, and it's all in Japanese! I will have to take a picture and post it for someone who knows Japanese to translate what it says on the tube!

But had to share for people to get the correct sliding pin lubricating grease. Hope this helps everyone who was looking for the correct grease!