“Update available” shown before Homebrew cask is updated, causing misleading upgrade message

What version of Codex is running?

0.56.0

What subscription do you have?

Team

Which model were you using?

_No response_

What platform is your computer?

MacOS(Darwin 25.0.0 arm64 arm)

What issue are you seeing?

When Codex shows the following update prompt:
✨ Update available! 0.56.0 -> 0.57.0

› 1. Update now (runs brew upgrade --cask codex)

1. Skip
2. Skip until next version

and I choose Update now, it runs the command successfully but outputs:

Updating Codex via brew upgrade --cask codex...
Warning: Not upgrading codex, the latest version is already installed

🎉 Update ran successfully! Please restart Codex.

This happens every time a new version is released on GitHub within the first few hours of the release.
It seems Codex detects the new version from GitHub Releases before the corresponding Homebrew cask has been updated.
At the time of filing this issue, the latest release (0.57.0) is already published on GitHub,
but the Homebrew cask has not yet been updated.

What steps can reproduce the bug?

1. Install Codex via Homebrew cask:

brew install --cask codex

1. Wait for a new version release on GitHub.

1. Launch Codex within a few hours after the release.

1. When prompted with “Update available!”, choose Update now.

1. Observe that Homebrew says “the latest version is already installed.”

What is the expected behavior?

The "Update available" prompt should only appear after the corresponding Homebrew cask has been updated and published

Additional information

Homebrew version: 4.6.20

It might be possible for Codex to verify that the Homebrew cask has been updated
(e.g. by checking https://formulae.brew.sh/api/cask/codex.json)
before showing the “Update available” prompt. This would prevent the prompt
from appearing too early, before the new version is actually installable via Homebrew.