This was not true for me, unfortunately. First night I played it I got through 50 of the fights, got all the tattoos, did the 10 minutes on the bag and probably 5 minutes on the weights, went to the quit option in the menu, and when I booted it up the next day it put me back at the beginning of the scene, before even doing the retrieve mission in the boiler room. All my tattoos and money from the fights the previous night were gone. I'd recommend anyone doing this to get to the next point you see the game say "saving" just to be safe. You can always finish the fights at a later point (chapter 7 according to people's posts in a different thread).
Thanks to jtr1gg3r for the awesome sig