If Toyota is gonna pay for it, ask for a brand new engine!
If it's your bill, I would suggest getting the motor out of the car and look at it. Maybe the rings in that cylinder are toast, but everything else is fine. In that case, it could be real cheap if you are willing to do the labor. You could resue everything and just replace bearings and rings (I highly recommend Total Seal rings), and do a rehone. Voila! Better than new for less than $200 in parts.
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As for the reason, I have two theories, depending on cylinder:
#1: Fuel rail design. After having a stock rail off the car to look at, I notice the regulator is between the #1 and #2 injector. Fuel does not go all the way across the rail. Rather it returns before getting to the #1. This could cause a fuel starvation problem on the #1 injector.
#6: EGR. Most of the hot gasses introduced into the intake by the EGR system find their way into the #6 cylinder (due to the EGR being in the very rear of the intake). If the EGR propped open during full boost, it could get very hot in the #6 cylinder and melt things.
These are just theories of course. Take them as you will. The first reason has me concerned, as I was hoping to keep my stock fuel system and inject tons of methanol to augment the stock system. This will not offer total protection from a #1 meltdown if my stock #1 injector only injects half of what all the others are injecting.