[Feature Request][Windows][Pets] Add a system-tray toggle to show/hide the desktop Pet

What variant of Codex are you using?

Codex Desktop app on Windows.

What feature would you like to see?

Please add a system-tray control for showing and hiding the desktop Pet without having to focus Codex and type /pet each time.

Problem

The Pet is useful as a lightweight status monitor while Codex is running tasks in the background, but the current toggle flow is too cumbersome for frequent use. When I want to temporarily hide the Pet and later bring it back, typing /pet in the Codex composer every time interrupts the workflow.

This is especially noticeable when Codex is running a longer task while I am working in another application such as Word, Access, or a browser.

Requested behavior

On Windows, when Codex is resident in the system tray, provide a quick Pet toggle such as:

• Left-click tray icon: show/hide (wake/tuck away) the Pet, or
• Tray context menu: Show Pet / Hide Pet (or Wake Pet / Tuck Away Pet).

If a single click on the tray icon is already reserved for opening the main Codex window, the context-menu toggle alone would still solve the problem.

Acceptance criteria

1. The Pet can be shown or hidden directly from the Windows system tray.
2. The main Codex window does not need to be opened or focused.
3. No /pet command needs to be typed.
4. Hiding and showing the Pet preserves the selected Pet and its saved position.
5. The control is available while Codex continues running background tasks.

Why this matters

The Pet is most useful when Codex is working in the background. A one-click tray toggle would make it practical to use as an on-demand task-status indicator without adding friction to the user's main desktop workflow.

Start by locating the Windows system-tray entry point and the existing /pet show/hide implementation in the Codex Desktop app. Trace how the selected Pet and saved position are retained, then verify the tray control works while background tasks continue and the main window remains unfocused.