I have offered this theory before, and nobody saluted. I will offer it again, but I don't expect any salutes this time either.
Air is compressible and fluids essentially are not. If you hit a bump of the proper shape under the right circumstances, perhaps that could be problematic if the tire is filled. Problematic to what? The tire, the axle, the bearings, who really knows. Fill the tires, put on the backhoe, and hit the right speed bump at the right speed, and send in the video and we will all have our opinion I reckon.
When I was looking at possible backhoe additions to my BX2200, the backhoes were well beyond what the lift could handle, which is ONE of the reasons they were sub-frame mounted.
My BX2200 has no backhoe, I decided to buy an additional tractor. But my BX would be nothing without filled tires. Were it me, I would fill the tires, but I would be a truly slow mover with the backhoe on.
The tires are not filled all the way so there is a cushion. They get filled to 75%.