Feature request: Make Codex side chats persistent first-class conversations

Problem

Codex side chats are useful for detailed analysis and follow-up discussion, but their lifecycle is not equivalent to a normal Codex conversation. Users can invest meaningful context in a side chat and then lose access to it after leaving the parent chat or restarting the app.

Related reports include #40262 (side chats cannot reliably be reopened) and #22001 (side-chat state is not persisted for later operations). This issue focuses on the product capability and user experience: side chats should be persistable by design.

Proposed behavior

Treat each side chat as a persistent child conversation of its parent chat:

• Persist the side chat and its messages across Codex app restarts.
• Show side chats in the parent chat and/or a dedicated history section.
• Give each side chat a stable ID and deep link.
• Allow users to rename, archive, fork, and delete a side chat.
• Provide a clear “Promote to regular chat” action when the user wants to continue it independently.
• Preserve the parent-child relationship without making the side chat inaccessible if the parent chat is closed.
• Where the associated Codex conversation is available, make persistent side chats reopenable from supported remote/mobile surfaces as well.

Acceptance criteria

1. Start a side chat from a parent Codex conversation and exchange messages.
2. Quit and relaunch Codex.
3. Reopen the parent conversation and find the side chat in its history.
4. Open the side chat and continue the conversation without losing context.
5. If persistence is not desired for a particular side chat, make that choice explicit and provide a clear save/promote option before the side chat is discarded.

Why this matters

Side chat should be safe for useful technical discussion. Without persistence or an explicit save workflow, users cannot tell whether important decisions and analysis will remain available later, which discourages using the feature.

Start by tracing how side chats are created, closed, and restored in Codex, then compare that lifecycle with persistent regular conversations. Use the acceptance criteria to define the finished behavior: relaunching Codex must preserve, rediscover, reopen, and continue a side chat without losing context.