tmux: variables sharing between sessions

I've noticed that

1. tmux will share variable between sessions.


2. tmux will increment TMUX_PANE for new session as well.

Is it possible to reset VAR for each session?

Is it possible to start TMUX_PANE from 0 for each session?

    # set variable
    # create 2 sessions
    $ export VAR=aaaaa
    $ tmux new -A -s $VAR -d

    $ export VAR=bbbbb
    $ tmux new -A -s $VAR -d

    $ tmux ls
    aaaaa: 1 windows (created Fri Jan 23 12:12:46 2015) [207x50]
    bbbbb: 1 windows (created Fri Jan 23 12:12:55 2015) [207x50]

    # connect to sessions
    # check value of variable
    $ tmux att -t aaaaa
    $ echo $VAR
    aaaaa

    $ tmux att -t bbbbb
    $ echo $VAR
    aaaaa            <-------- i would like to see 'bbbbb' here