[Feature Request] Show desktop pet only while a Codex task is running

What variant of Codex are you using?

App (ChatGPT desktop app, Codex)

What feature would you like to see?

Please add an opt-in desktop pet visibility mode that shows the floating pet only while at least one Codex task is actively running.

Suggested setting under Settings > Pets:

Show pet only while a task is running

Expected behavior when enabled:

• Keep the pet tucked away when no task has the Running status.
• Automatically wake/show the pet when any Codex task enters Running.
• Keep it visible while at least one task remains Running, including when Codex is not the focused app.
• Automatically tuck it away when no tasks remain Running.
• Preserve the selected pet and its last position between runs.
• Continue to allow the existing /pet, Wake Pet, and Tuck Away Pet controls to override visibility for the current run if practical.

Today, the pet can be woken or tucked away manually, and it already understands the Running status, but it remains as an always-on-top overlay while idle. I would like to use it as a lightweight activity indicator without keeping it on screen when Codex is not doing work.

This would make the pet useful as an ambient “Codex is working” signal while avoiding persistent desktop clutter and distraction between tasks.

Additional information

• ChatGPT desktop app: 26.818.61809 (build 7019)
• Platform: macOS 26.6.2 (25G83), Apple Silicon
• Current workaround: manually run /pet before and after each task.
• The current pet documentation describes manual Wake/Tuck Away controls and the Running status, but no automatic visibility mode.
• I searched existing openai/codex pet issues. Related requests cover hiding the pet during fullscreen/screen sharing (#33517) and completely disabling Pets (#34349), but I did not find this running-only lifecycle request.