There is no "deep" to ratfur!
but there is a product which cleans it really great without harm.
I have used Shout Carpet Cleaner for spot cleaning with great success and little effort. They also make a concentrate for general cleaning.
The package shown below comes with a built-in brush dispenser.
I have no personal experience with STEAM but I am concerned about its effect on reflextix and the adhesives. Heck, it could release the wall covering or reconstitute it... proceed with caution!
Maybe one of those little steam cleaners made for curtains and clothing might work to freshen up and general cleaning after spot cleaning with the Shout, but there are plenty of automotive products which would work well such as Scothguard products...