Wrong session_info in nested tmux session

[happyman]

I have a 6 tmux sessions named p1~p6, and the following script to create a group to show p1 to p6 at once.

#!/bin/bash

tt="tmux att -t"
SESSION=test
tmux kill-session -t $SESSION
tmux new-session -t $SESSION -d
tmux rename-session -t $SESSION $SESSION
tmux ls

tmux split-window -h -p 50 -t $SESSION:0
tmux split-window -v -p 66 -t $SESSION:0
tmux split-window -v -p 33 -t $SESSION:0
tmux select-pane -t $SESSION:0.0
tmux split-window -v -p 66 -t $SESSION:0
tmux split-window -v -p 33 -t $SESSION:0
tmux move-window -r -t $SESSION:0
tmux ls

tmux send-keys -t $SESSION:0.0  "TMUX='' $tt p1" C-m
tmux send-keys -t $SESSION:0.1  "TMUX='' $tt p2" C-m
tmux send-keys -t $SESSION:0.2  "TMUX='' $tt p3" C-m
tmux send-keys -t $SESSION:0.3  "TMUX='' $tt p4" C-m
tmux send-keys -t $SESSION:0.4  "TMUX='' $tt p5" C-m
tmux send-keys -t $SESSION:0.5  "TMUX='' $tt p6" C-m

tmux att -t $SESSION

but the session info will be the last atteched one.
for example, if I attach to p2 within session test, then all session names displayed in status bar will become p2.

is there anyway to fix it?