I rode our system between Memphis and a couple of domestic outstations this summer with my road bike in a Thule hard case... no issues. Security will have you open up your bike case so they can inspect everything.
The prohibited items list mentions that bicycles are required to be loaded into the ABK or a compartment that is inaccessible during flight. It's also got some weird guidance for certain kinds of tools that you may be carrying in order to assemble the bike at your final destination.
Don't forget to remove any CO2 cartridges if you carry those for flat repairs.