I have an Epiphone with humbuckers. Just put new pickups in it. Two volumes and one spin-a-split (coil tap using a pot) for the bridge pickup. No tones and no other pot wired to the neck. Also a 3 way. I have buzzing that goes away when I touch the cavity paint or the jack. All components are good. All soldering points aer good as well.
I think it is a grounding problem.
please see the attached paint file. It shows what I currently have done.