Default global new chats to No Project instead of the last active project

What variant of Codex are you using?

Codex Desktop / Codex App on macOS.

What feature would you like to see?

The global new-chat composer should default the project selector to No project / General instead of preselecting the last active project.

Only chats opened explicitly from a project context, such as a project sidebar/page or a project-specific "new chat" action, should preselect that project.

Why this matters

A new chat often does not pertain to any repo or project. It may be a general question, GitHub/account/admin task, browser task, planning task, product question, debugging triage, or a prompt where the correct project is not known yet.

When the composer silently defaults to the last active project, it is easy to start a thread in the wrong repo. For a coding agent, that is a risky default because project context affects search, files, shell cwd, permissions, memory, and the user's mental model of where the work is happening.

Expected behavior

• Global new-chat entrypoint defaults to No project / General.
• Project-scoped new-chat entrypoints default to that specific project.
• The dropdown includes an explicit No project option.
• If preserving the current behavior is useful for some users, expose it as an opt-in setting such as Default new chats to last active project.

Observed behavior

The new-chat project dropdown can default to a previous project/repo, requiring manual switching back to no-project mode before starting unrelated chats.

In local state this appears to correspond to an active project selection such as active-workspace-roots, but the product issue is the UX default: a global new chat should not inherit project scope unless the user intentionally starts from that project.

Related issue

Related to #24222, which covers the broader General/Project thread context model and moving threads between contexts. This issue is the narrower request for the default selection in the global new-chat composer.