I'm trying to use sudoedit, but it keeps running the nano editor. My preferred editor is vim. How can I make it the default?

Looking at the man page, man sudoedit, I've noticed the following:

 EDITOR           Default editor to use in -e (sudoedit) mode if neither SUDO_EDITOR nor VISUAL is set.

 SUDO_EDITOR      Default editor to use in -e (sudoedit) mode.

 VISUAL           Default editor to use in -e (sudoedit) mode if SUDO_EDITOR is not set.

So I've set them all to /usr/bin/vim, but sudoedit /etc/hosts still uses nano. Am I missing something?

$ EDITOR=/usr/bin/vim

$ VISUAL=/usr/bin/vim

$ SUDO_EDITOR=/usr/bin/vim



$ echo $VISUAL

/usr/bin/vim



$ echo $EDITOR 

/usr/bin/vim



$ echo $SUDO_EDITOR 

/usr/bin/vim



$ sudoedit /etc/hosts # This is still using nano