Codex Desktop should show current project, cwd, and Git branch in thread UI

Summary

Codex Desktop should show the active project/workspace directory and Git branch prominently in the thread UI, similar to how the CLI can expose project and branch context in its status line.

Problem

In Codex Desktop, it is not obvious which local project/directory a thread is currently operating in, nor which Git branch or worktree is active. Users can ask Codex or open the integrated terminal and run pwd, git rev-parse --show-toplevel, and git branch --show-current, but that is friction for something that is fundamental context for every coding task.

This is especially confusing when:

• multiple Codex Desktop threads are open across different projects;
• a user switches between Local and Worktree modes;
• projectless chats create Codex-managed temporary directories;
• a user expects parity with the CLI/TUI, where status-line context can be configured;
• a task involves commits, reviews, deploys, or any operation where the wrong repo/branch is costly.

Requested UX

Add a persistent, visible project context indicator to Codex Desktop threads.

Suggested fields:

• Project name
• Current working directory
• Git repository root, when different from the working directory
• Current Git branch
• Local vs Worktree vs Cloud mode
• Worktree path/name, when applicable
• Dirty state indicator, if cheap and reliable

Suggested behavior:

• Show a compact label in the thread header, for example: my-repo · feature/foo · Worktree.
• Allow clicking/hovering to reveal full paths.
• Provide quick actions to copy cwd, copy repo root, open terminal, open Finder, or open the project in the configured editor.
• For non-Git/projectless chats, show that explicitly, for example: Projectless chat · Codex-managed directory.
• Keep this visible during long-running work and when approval dialogs appear.

Why this matters

This context is safety-critical, not just cosmetic. Codex can edit files, run commands, commit, push, create worktrees, and operate across multiple local projects. Users need continuous awareness of where the agent is operating before approving commands or interpreting results.

The CLI already has configurable status-line concepts such as project and git branch. Desktop should provide an equivalent first-class affordance rather than requiring terminal commands or asking the agent.

Current workaround

Open the integrated terminal and run:

pwd
git rev-parse --show-toplevel
git branch --show-current

or ask Codex to run those commands. This works, but it is not discoverable and does not provide persistent situational awareness.