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We have an '06 3500RL which is similar to the '06 3400RL (just a different floorplan). The GVWR is 14,040, the axles are 6,000 rated meaning the remaining 2,040 is intended to be carried at the pin.
The real life weights for my rig are 13,500 Gross weight, 2,600lbs at the pin leaving 10,900lbs over the axles. Note all numbers fit the axle ratings just fine. This all changes when the tanks are full, therefore I ensure my tanks are virtually empty before getting on the road.
Since your rig has more stuff like the 2nd A/C, and a generator, you can't carry as much junk as I do since you are probably closer to your GVWR of 13,960 (80 lbs less than my rig). I would guess the rig label is correct that you have 6,000 lb axles.
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