Can trigger spell skill get bonus level from wand/staff?

I thought the ''+4 level of all cold spell skills'' in the wand/staff is global, but when I checked the Comet socketed in my 'Cast on...', it didn't have that bonus and only got the level bonus from the support gem.

Does anyone know anything about this?

It doesn't really make sense to me but apparently +skill levels only apply to spells that are actively cast.