display-menu vs run-shell vs new-window

[097115]

Let's say I have the following script:

#!/usr/bin/env bash

tmux set-env -gu TMUX_TEST

menu=$(echo "\"aaa\" \"1\" \"setenv -g TMUX_TEST aaa\" \
             \"bbb\" \"2\" \"setenv -g TMUX_TEST bbb\" \
             \"ccc\" \"3\" \"setenv -g TMUX_TEST ccc\"")

eval tmux display-menu "${menu}"

TM_TS=$(tmux showenv -g TMUX_TEST 2>/dev/null) || exit
TM_TS=${TM_TS##*=}

tmux display-message $TM_TS

1. Now, if I run it from the command line tmux run-shell PATH/TO/script, everything works fine.
2. However, if I call it from another menu tmux display-menu "script" "1" "run-shell PATH/TO/script", it fails.
3. And if I replace the latter run-shell to new-window, it works again: tmux display-menu "script" "1" "new-window \"PATH/TO/script ; read -n1 q\"", it works again.

Why is that? And is it possible somehow to fix the 2. scenario?