The white wire is a ground. It grounds the solenoid so that the solenoid then gives power to the GPs.
When you do that, the two yellow wires start to melt. That is like taking those two yellow wires and putting them straight to ground. They carry full battery voltage. Take those off of the solenoid, and directly ground them back to the battery, and they will melt like that.
This ONLY happens when you switch the solenoid. Which then connects the yellow wire post, with the other main post.
It's possible that your GP harness is grounded straight to ground somewhere. Not just feeding the glow plugs. That would essentially be the same as running the yellow wires straight to ground.
It's possible the solenoid is grounded out somehow. I don't know how.
It's possible the ribbon is grounded. Maybe there is a problem inside the controller? The bolt that goes through the other end of the ribbon (opposite end form the solenoid), also goes down into the controller. Maybe it has burned through something down there and grounded itself out?
I'd think your next step is to remove the controller from the truck and examine it thoroughly. Look for signs of burning.
Gotta say, this is part of why I am going to make my own GP solenoid circuit. My controller or solenoid just recently went out. GPs are not working. Before it went out, my voltage meter on the dash was acting wonky. After it went out, my voltage meter is back to normal.
This makes me think the GP controller is a risk. Specifically a fire risk. And your experience only strengthens that suspicion.
And I especially dislike that all of that is buried down behind the intake and under the air cleaner. When I build mine, I am putting it right out in the open where I can see it and get at it.