Nor have I any sure information about composting corrugated cardboard, though I have used a great deal of it in the landscape, and in my home composting operation.
I think it would be perfectly OK to use as you describe at the base of trees. I might be inclined to soak it in water first, to get it really saturated. Then cover it with wood chips. I doubt that 2" is enough depth to keep the cardboard out of view, though. I'd go for 4", but I really like wood chips. :)
I did use corrugated cardboard to eliminate a very weedy lawn. The only downside was that my property has a great deal of equisetum (horsetail) which enjoyed spreading its roots into the spaces left by the wavey layer in the middle.
I have a pretty active composting operation in my backyard and I have often thought that dishwasher or clothes washer boxes would serve well as temporary composters - just something to keep the raccoons and possums from dragging the compost around the yard. I've not tried that, though. I have a plastic composter that is missing the door at the bottom. I use a flattened corrugated box to cover the hole and keep compostables from spilling out. It's remarkable how long (months and months) that box will serve.
If my goal were to compost cardboard - it would have to be one component of various materials to be composted - it would have to be broken up into many more pieces. The point here is not to reduce the size so much as to create many more feet of raw edges where water could be absorbed.
As it is, I do compost a good deal of shredded office paper. It's absorbent, and the chemical processing that turns wood pulp into paper has already made the material faster to break down - usually within a week in an active compost pile, paper is no longer recognizable as paper. Cardboard is certainly thicker and less bleached that writing paper. Both of those characteristics would make it slower to break down. I only use waxed cardboard in the landscape if I want something that will really last.