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I assume you've tried a different cable to make sure that it's not the cable, and a different guitar to make sure it's not the amp? Okay.
My wild but somewhat educated guess: the jack. I have a 2000 (or thereabouts) Epi Firebird. One of the less impressive qualities of it is the electronics. The pickups were just okay, but the jack, switch, and pots were poor, and the wiring resembled a rat's nest. But I'll assume that you just want to fix the immediate problem for now, not redo everything.
Judging by the fact that both pickups are equally affected, one would suspect whatever part of the wiring is common to both, and I'd suspect something that is merely metal-to-metal first, rather than something soldered. For some reason, the metal contact surfaces of the jack and switches on some of these Epis seem to tarnish and provide poor contact after a relatively short time.
I'd recommend changing out the jack for a Switchcraft or Gibson part. It's a pretty cheap thing to buy, doesn't take long to do, and if you still have the problem, well, you would have wanted to change it eventually, anyway.
The technically correct method would be to troubleshoot it with a meter, but that's rather 'fiddley,' takes a long time if you're not experienced, and there's no guarantee that contacting parts of the jack with the meter probes, in this case, are going to be the same as the cable plug contacting the jack, anyway. I'd just change the jack, and if it is still bad, then go ahead and troubleshoot with a meter.
It was the switch in my Firebird that started giving me grief. I took the damn thing apart and tried to clean it a bunch of different ways, a bunch of different times, but it just kept screwing up soon after I put it all back together.
I finally just redid all the electronics like they would be done in a Gibson, replacing everything with USA parts and cable, and it's been electronically solid, and sounding considerably better, ever since.
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