sudo не подставляет $PATH целевого юзера

-i [command]
The -i (simulate initial login) option runs the shell specified in the passwd(5) entry of the target user as a login shell.  This means that login-specific resource files such as .profile or .login will be read by the shell.  If a command is specified, it is passed to the shell for execution.  Otherwise, an interactive shell is executed.  sudo attempts to change to that user's home directory before running the shell.  It also initializes the environment, leaving DISPLAY and TERM unchanged, setting HOME, MAIL, SHELL, USER, LOGNAME, and PATH, as well as the contents of /etc/environment on Linux and AIX systems.  All other environment variables are removed.

user@host:~$ env | grep test
user@host:~$ sudo su -
root@host:~# echo 'export PATH="$PATH:/tmp/test"' >> /root/.profile 
root@host:~# source /root/.profile
root@host:~# env | grep test
PATH=/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/etc/alternatives/gem-bin:/tmp/test
root@host:~# logout
user@host:~$ env | grep test
user@host:~$ sudo env | grep test
user@host:~$ sudo -i env | grep test
PATH=/usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/games:/etc/alternatives/gem-bin:/tmp/test:/tmp/test
user@host:~$