A terminal command that starts a GUI app (flatpak run, google-chrome,
xdg-open) inherited the gateway's own DISPLAY, so while a Bot Screen was
running the window opened on the user's seat and xrandr/xdotool probed
the user's display — the exact babysitting loop of #125830. Merge the
launcher's published DISPLAY/XAUTHORITY/DBUS into _make_run_env while
the Bot Desktop is up (same routing the browser tool and cua backend
already apply), via published_env() so a plain command does not stamp
screen activity and idle_stop_minutes stays effective. An inline
DISPLAY=:N prefix still wins; with the desktop down the seat env passes
through untouched.