[BUG] Side chat (Cmd+;) stops responding after the first reply — follow-ups produce no output

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

### Preflight Checklist
- [x] I have searched existing issues and this hasn't been reported yet
- [x] This is a single bug report
- [x] I am using the latest version of Claude Code

### What's Wrong?

In the Code tab, opening a side chat (Cmd+; or /btw) and asking a question
returns a normal reply. Sending anything into the "Follow up…" input after
that produces no output — no response, no error, no visible state change.
The panel stays open and accepts input, but nothing comes back.

The first reply also frequently ends by promising a continuation
("Let me show it with one of your actual campaigns…"), which the follow-up
turn never delivers.

Two possibilities, and either one is worth fixing:

1. Multi-turn side chat is intended and is broken.
2. Side chat is intentionally single-turn (matching the original /btw
design: one turn of output, no tool access, full conversation context).
In that case the persistent "Follow up…" input and the follow-up
affordance in the panel are misleading UI — they invite a second turn
the feature can't serve.

The desktop docs (code.claude.com/docs/en/desktop, "Ask a side question
without derailing the session") don't state a turn limit either way, so
there's no way for a user to tell which of the above is happening.

### Claude Model
Opus 5

### Is this a regression?
Unsure — I believed follow-ups worked previously, but I can't confirm a
last-working version.

### Last Working Version
No response

### Claude Code Version
1.46388.1

### Platform
Claude Desktop app, Code tab, local session

### Operating System
macOS Tahoe 26.6.2

### Terminal/Shell
N/A (desktop app)

### What Should Happen?

Either:
- Follow-up messages in a side chat get a response using the same
snapshotted session context, or
- The side chat disables/removes the follow-up input after the first reply
and states that it's a single-turn feature, and the docs say so.

### Error Messages/Logs

```shell
None surfaced in the UI.
```

### Steps to Reproduce

1. Open a local session in the Code tab with meaningful conversation history.
2. Press Cmd+; to open a side chat.
3. Ask any question that draws on the main thread's context.
4. Wait for the reply to complete.
5. Type anything into the "Follow up…" input and press Enter.
6. Observe: no response is produced.

### Claude Model

Opus

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

1.46388.1

### Platform

Other

### Operating System

macOS

### Terminal/Shell

Other

### Additional Information

Desktop app version: 1.46388.1
Session type: local (not SSH, WSL, or cloud). Main window, not a detached
window — this is distinct from #82924, which covers side chat being
unavailable in detached windows on Windows.
standalone cli version: 2.1.251 (Claude Code)

Start by reproducing in the Claude Desktop app Code tab on macOS 1.46388.1 using Cmd+; or /btw, then sending a follow-up after the first side-chat reply. The issue names no files or tests, so first locate the side chat UI and follow-up handling entry points. Done means follow-ups either produce responses or the UI/docs clearly state and enforce that side chat is single-turn.

Do mô hình lập chỉ mục viết ra từ nội dung của issue.