Nope, didn't work.
I have now tried to use one of the shell accounts to ssh into the other, since I don't have any working Unix system at home. So I generated a private key, moved it to my other users id_rsa file, moved the public key to my own authorized_keys file, then su'd to the other user and tried to ssh back to the first user.
The output using -v is this:
debug1: Authentications that can continue: publickey,password,keyboard-interactive
debug1: Next authentication method: publickey
debug1: Trying private key: /home/USER/.ssh/identity
debug1: Trying private key: /home/USER/.ssh/id_rsa
Enter passphrase for key '/home/USER/.ssh/id_rsa':
debug1: Trying private key: /home/USER/.ssh/id_dsa
debug1: Next authentication method: keyboard-interactive
Edit: Note that it does check the passphrase properly, so the last three lines of output only occur when the correct passphrase is entered.