Add side-chat-specific default model and reasoning settings

Summary

Please add a way to pin model and reasoning-effort defaults specifically for Codex side chats, independent of the main task thread.

Problem

Codex side chats are useful for asking quick clarifying questions, status checks, summaries, or "what is going on here?" explanations without interrupting the main agent task.

Right now, side chats appear to inherit the active/default Codex model and reasoning settings. That is not always ideal. A main coding task may need a stronger model and higher reasoning effort, while side chats are often lightweight and should be fast, cheap, and low-friction.

For example:

• Main task: gpt-5.5 with high or xhigh reasoning.
• Side chats: a faster model, or the same model with minimal/low reasoning.

Without a side-chat-specific default, users have to manually adjust the model or reasoning each time, or accept that quick side questions consume the same heavier configuration as the main task.

Requested Behavior

Add side-chat-specific defaults in Codex configuration.

Possible config.toml shape:

[side_chats]
model = "gpt-5.4-mini"
model_reasoning_effort = "low"

Or, using top-level keys:

side_chat_model = "gpt-5.4-mini"
side_chat_model_reasoning_effort = "low"

When a side chat starts, Codex should use these values if present. If they are not present, current inheritance behavior can remain unchanged.

Why This Helps

This would let users optimize main task and side-chat behavior separately:

• Keep the main agent on a high-quality, high-reasoning setup for implementation work.
• Keep side chats quick and inexpensive for status questions, explanations, and navigation.
• Reduce accidental token and credit usage from lightweight side conversations.
• Make side chats feel like a low-cost "ask the agent what is happening" channel.

Expected UX

Side chats should show the selected/default side-chat model and reasoning setting clearly, just like normal chats do.

Users should still be able to override the model or reasoning setting manually inside a specific side chat. The configured value should only be the starting default.

Non-Goals

This request is not asking to change the main thread's model behavior.

It is also not asking to force all side chats to use a fixed model if a user manually changes it inside the side chat.

Related Issue

This is adjacent to #31226, which asks for mode-specific defaults for Standard chat, Plan mode, and Pursue Goal mode. This request is narrower and specifically about side chats, which are a distinct interaction pattern during an active task.

Suggested Priority

Medium. This is not a correctness bug, but it affects day-to-day ergonomics, cost control, and how comfortable it feels to use side chats freely during long-running Codex work.