I love these types. This is the kind of thing that companies call me in for at great expense to themselves.
Unfortunately, this one sounds simple. Not much of a challenge.
First thing we have to do is isolate the mechanism that causes it.
Replace the fuse, turn the key on and press the accelerator. Does it go up?
Crank the engine, leave it in park or neutral and press the accelerator. Does it go up?
I'm betting that it does not.
If not, then you can forget about anything to do with the accelerator its self. The problem is caused by acceleration. More specifically, inertia.
As you accelerate forward, inertia causes everything to want to move back.
This movement is what triggers the window.
If I remember right, the window is normally triggered by a key switch in the back. Is there one on the dash also?
I would turn on the key, so the window has power, and then either push from inside, or pull from outside, the back door to simulate acceleration.
In this test, I am thinking you have a shorted switch and the power contact only makes contact when the door moves.
If not, then probably there is something in the switch mechanism itself that is making contact when you move forward.
Something could be loose in the door and swinging up against the switch contacts.
The switch contact inside the switch will wear out and break sometimes, and it might be moving around inside the switch and causing this.
Bear in mind, I have not seen one of these systems, so some of these ideas may not be possible.
I could do a lot better in person.
But if you can isolate the basic mechanism with the first tests, it will go a long way to tracking it down.