Well, the flux density inside the iron of the tube is certainly greater than elsewhere, but that does not reduce the field outside the tube.
Magnetic shielding is different. Ampere's law really works like that. Moreover, whereas a thin metal box can stop an electric field, you need a thick-walled shield to guide a magnetic field around your critical volume. Oddly enough, the shield does not need to be closed. (Well strictly, that is true for E-field shields too.)
Around 2006, I installed a box with little RF transformers in the fringing field of the bending magnets of the Proton Synchrotron here. The transformers were inside a little pill-box style mu-metal shield with 1mm wall thickness. Near 10GeV, my transformers suddenly stopped working. The shield was not good enough!
I solved the problem by sandwiching the works between two 5mm thick soft iron discs, taking care to align the discs' diameter with the field. That way, the flux goes through the iron and little is left in between the discs.
Jeroen Belleman