How to get the name of the monitored session in the custom command?

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I have many sessions on my server, so I want to get the session name of the monitored session and attach it to the notification email. I tried tmux display-message -p '#S' (https://superuser.com/a/410197/1677998):

set -g @tnotify-custom-cmd 'echo $(tmux display-message -p \#S) finishes" | s-nail --subject="tmux complete notification" my@email.com'

However, it seems that the attached session is the last attached session instead of the monitored session. Then I tried to get the tty name of the monitored session and find the corresponding session (https://superuser.com/a/1202709/1677998):

# ~/session-name.sh

tty=$(tty)
#echo $tty
for s in $(tmux list-sessions -F '#{session_name}' 2>/dev/null); do
    tmux list-panes -F '#{pane_tty} #{session_name}' -t "$s"
done | grep "$tty " | awk '{print $2}'

set -g @tnotify-custom-cmd 'echo "\"$(bash $HOME/session-name.sh)\" finishes" | s-nail --subject="tmux complete notification" my@email.com'

However, tty=$(tty) returns a not a tty result.

I also tried https://superuser.com/a/1357132/1677998:

if [[ -n "$TMUX_PANE" ]]; then
    session_name=$(tmux list-panes -t "$TMUX_PANE" -F '#S' | head -n1)
fi

However, TMUX_PANE is not defined.

Is there any way to get the session name of the monitored session in the custom command? I need this because attaching up to ten sessions one by one to find the one that finishes is very painful.