What ego? :3 I see no ego here!
[Well, aside from the tank from his recollection, around his neck most often. At least most recently...
Yes, the TARDIS has something for it, it's just... a bloke likes his clothes professionally cleaned by a proper dry cleaner sometimes. Oh and one or two of the parts for the machine that does it may be missing due to a console repair a few months back. :/ Dry cleaning it is then.]
Safe keeping? I'd have stuffed it in a pocket before shoving it into a toilet's water reserve tank! [And he's wringing the poor thing out over the sink. S-so sad.