Use the grease Campy was thoughtful enough to provide for your initial installation. When the headset is due for an overhaul in a couple of years, wipe off the then dirty grease thoroughly and replace it with whatever type you choose. Since you already have Phils, use it then and you can't go wrong. I use Phils for all my bike bearings and have had great parts longevity with it.

Grease tend to be badly overthought and overargued here. Just about any decent grease will do fine for bike use since the demands are very low. "Over priced bike specific grease" isn't required but it works well and the extra cost is peanuts in the long run.