Unmount user home directory in a delayed background process after logging out

The home directory of user bob is mounted by a superuser from elsewhere. bob himself is a sudoer, he is logged in and is happily using his mounted home directory. He finishes his work and wants to run a script which will log him out *and* unmount his home directory. He tries:

cd / # switch to a directory other than /home/bob to avoid keeping it busy
bash -c 'sleep 5 && sudo umount /home/bob /dev/null & disown
gnome-session-quit --no-prompt

He gets logged out but then, when alice logs in (also a sudoer), she sees bob's home directory still mounted after a while. Apparently, the umount command fails for some reason. How can bob reliably unmount it upon logging out?