Ground issues can be confusing, I figured the basic theory would be worth mentioning - again, more for the uninitiated... sometimes those ground lugs are also for common ground power distribution via the chassis.
Also worth mentioning... the P6 uses a grounded plug, which occasionally can be an issue when connecting it to external devices with grounds or leaky AC, like mixers, computer interfaces (USB midi on the korg SQ1) and such, which result in odd noises ... but I think that was covered by reverting back to the headphones being the only device plugged into the P6?
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The foil is already either acting like an antenna or creating a ground loop. Since the “real” ground is connected, it is probably a ground loop, so eliminating that one ground could cause a significant difference. Either way, it can provide some clue to what is going on if it makes a difference.