Control+Space still toggles Pet and conflicts with macOS input-source switching

What variant of Codex are you using?

ChatGPT desktop app, Codex surface

• App version: 26.901.51231
• Build: 8109
• macOS: 26.6.2 (25G83)
• Architecture: Apple Silicon (arm64)

What happened?

Pressing Control + Space to switch between Chinese and English input methods also toggles the floating pet. The pet repeatedly appears and disappears during normal typing, which interrupts the user's workflow.

This appears related to #42726, which was closed after being reported as fixed in the newest update, but the conflict still reproduces on the version listed above.

Steps to reproduce

1. Keep the ChatGPT desktop app running.
2. Configure or use macOS Control + Space for “Select the previous input source.”
3. Press Control + Space while working in any app.
4. The input source changes, and the Codex pet also wakes or tucks away.

Expected behavior

Control + Space should continue to switch the macOS input source without toggling the pet. Please remove this global default binding, or provide a setting to disable or remap the pet shortcut.

The issue names no file, test, or entry point. Start by locating the macOS desktop-app code that registers the floating pet's global Control+Space shortcut, then reproduce the conflict with the listed macOS input-source setting. Done means input-source switching no longer toggles the pet, with any remapping or disable setting covered by the chosen behavior.