Have you tried playing that tone and shifting it's phase to cancel the imaginary signal?
I still think it's unlikely to work.
In simpler terms: tinnitus does not have phase, at least not in the same way a sound wave does.
Funny enough lots of stereo widening magic relies on a physical property of sound not existing when it's in your head.
I've never tried it with headphones, but I suppose even though destructive addition isn't going to happen it might affect things like intended spatial perception.
This technique makes the ringing go away for a while, but it creeps up again after some time. Thankfully, it has always completely gone away after a day or two.